moon hoax essays

moon trick papers That is one little advance for man, and one goliath jump for humankind, said Neil Armstrong when he originally set a stage on the moon. In any case, there have been some intriguing hypotheses about the moon arrival that question the U.S. in it's notable accomplishment. A show that broadcast on fox that was named Conspiracy Theory: Did we land on the moon? introduced proof that brought up alot of issues about the moon arrival. The fear inspired notion suggeted that the photos, the props, for example, the foundation and the a few realities about the atmospehere and the moon all add to what the Fox makers accept to be the main response to those faulty pictures and realities: it was every one of the a trick. NASA anyway has a clarification for those vulnerabilities and discloses thouroughly concerning why a few things lke the photos and entire landing is solid. In spite of the fact that the memorable accomplishment of the U.S.has been subverted, NASA gives adequate data about the moon ar rival and why it is valid. The flawed data that was There is some intriguing data about the arrival in any case, that makes the United States look exceptionally terrible most definitely. The data that was introduced disclosed on Fox around 3 years prior, and presented the possibility that the entire moon landing was a major fabrication. There are a few snippets of data, similar to pictures, that truly make individuals question the United States accomplishments in space. Fascinating realities about the moon propose that the excursion to the moon would be almost unimaginable. Likewise, a recommendation that the entirety of this organizing was accomplished for someones insatiability is additionally a chance. Cash, realities of the moon and a few pictures from the moon present a contention with believability and extreme to address addresses that truly question the accomplishment of the United States. Some intriguing photos of the moon present data that makes individuals, including myself, wo ... <!

Police Accountability Essay

Cop Accountability is arrangement practice greatest thing. An official is responsible for the network, the division, and themselves (Peak, 2012). An official activity is a responsibility which can welcome on more strain and worry that may hinder the officer’s split dynamic when an official is engaged with an interest, the official must remember the open wellbeing, just as the security of the suspect, and when he capture the speculate he should ensure that he submit to the law and to ensure that the purpose behind the interest is however in the report for court. The repercussions would ensure his responsibility is effective or assume liability if something awful occurs. A case of a particular circumstance where official responsibility for his activities that would influence his dynamic in authorizing the law would be: An official is sitting in a store parking area and he sees a vehicle speeding through the parking area traveling toward individuals who are going all through the store. The vehicle runs more than one of the individuals and continues onward. The official takes off behind the vehicle and pulls it over. The official gets of his vehicle and stroll to the driver side of the vehicle, as he approach the vehicle and begins to approach the driver for his permit and enlistment he notice that it is one of his individual official and he is very inebriated. The off the clock official requests that he let him go with a notice not understanding that he because pulled over was on the grounds that he just ran somebody over. In the event that the official who is on the job releases him he would be considered responsible in light of the fact that he would facing a challenge by letting him proceed to drive and hazard him hurting another person or himself. The official captures his individual official so he can be considered responsible for his unlawful activities.

Finding Funny Satire Essay Topics

Finding Funny Satire Essay TopicsComedy and satire are the most important tools in a successful satirical essay. A good example of this is David Sedaris' collection of essays entitled, 'The Editor's Favorite Weirdo' (1984). Sedaris would typically choose a topic that is funny or satirical enough to be taken seriously and then write a satirical essay based on that topic. It can often be difficult to avoid sounding like a caricature of yourself, so you must strive to sound like you're making a serious argument and not just rambling off nonsense.When making your essay, make sure that you first define your topic in order to make sure that it isn't too complicated for you to convey in writing. Too many people make their topics too complicated by just throwing too many words into the piece. If you're interested in a topic like philosophy, for example, do not try to cram the entire world into a few hundred words. Be specific and write about the topics you're interested in. Also, be sure to limit your topic to just a few things.Secondly, make sure that you are aware of the difference between satire and sarcasm. While humor, like anything else, can be used for both, a satire will use humor to put down a subject while a sarcasm will use humor to praise a subject. In order to effectively write a satirical essay, you'll need to make sure that you take the time to think about how other people will view your essay. As an example, a parody of a religious essay will likely be read as a joke rather than an essay that question the concept of God.Many ways exist to find interesting, funny satirical essay topics. One way is to search for them on the Internet. If you find something that you like, go online and look for other similar essays to see if you can find similarities and differences. Another thing to consider is that, most of the time, when you find a humorous essay, it was written fora specific purpose. You may be trying to find the humor, but they were also written for a specific purpose.The Internet is also a great place to find funny satirical essay topics and to read other essays written by others. This can be a good place to get ideas and to meet people who share your interests. You can also go to talk shows and comedy shows to see what topics are being discussed. This can be a great way to meet people that share your interest.Remember that while writing your funny satirical essay topics, make sure that you don't feel the need to be clever or original. Just write in a sarcastic manner, and if you do write something original, it should be in a non-satirical way. Of course, you want to be aware of the fact that your essay is not going to be accepted at places like college admissions. It's a serious matter, so don't try to have too much fun with it. People won't think that you are clever, so make sure you maintain your seriousness.An important lesson here is to try not to be so polished. Try writing using lots of different topics, because no one li kes to read just one kind of essay. However, don't be too scatterbrained either. You don't want to be too predictable or boring as well.Remember that you need to make your funny satirical essay topics relevant and sound thought provoking. While the whole point of an essay is to express an opinion or to convey an idea, you also need to make sure that you are consistent with your choices. You also need to make sure that you don't try to be clever in your choice of topics.

Consumerism in Post World War II Essay - 1479 Words

Consumerism in Post World War II After WWII why did the economy prosper and what role did consumerism play in the 1950s? After WWII many economists predicted a recession in the American economy. It is easy to do so when at the peak of post war unemployment in March 1946 2.7 million searched for work. In 1945 people were laid off from their jobs. However, â€Å" in 1945 the US entered one of its longest, steadiest, periods of growth and prosperity† (Norton 829). How could this be? With many new developments affecting the United State’s social and economic behavior, the wealth of the nation burgeoned. It is the extreme wealth of this society which supports and creates consumerism, the â€Å"Americans’ [increased] appetite for goods†¦show more content†¦Businesses need a â€Å"high level of economic security† in order to sustain maximum production (Galbraith 111). So the increase in the production was not to create more goods, although that resulted, but to secure more economic stability. The goods are secondary compared to the their â€Å"assured productio n means assured income for those who produce them† (Galbraith 114). This typical attitude towards production changes in the 1950s. Galbraith’s five ways in which production can be increased pertains to the 1950’s economy. His first method is to employ labor and capital more (Galbraith 119-120). In the 1950s industries started to invest more capital. In 1950 1.1 billion dollars of capital was invested by industries compared to 1.86 billion dollars in 1959 (Pate 669). Secondly, Galbraith mentions that labor and capital should be used in â€Å"the most advantageous combination, one with the other, and the two can be distributed to the greatest advantage, consumer tastes considered, between the production of various things and the rendering of various services† (Galbraith 120). The conglomerate mergers that took place in the 1950s exemplified this suggestion. A conglomerate joined companies in differing industries to combat instability in a particular market (Norton 831). The business corporation became an important part of the functioning of American life. Conglomerate firms represented the most prominent form ofShow MoreRelatedThe Post World War II1128 Words   |  5 PagesAfter World War II ended in 1945, many significant changes to American society began to occur. Some of these major changes helped shape what the U.S. is today and include the Baby boom, mass suburbanization, and mass consumerism. The Post-World War II era is defined by these changes in U.S history and culture. In this Post-World War II era, social conformity became the most ideal way of life. Every citizen wanted the same thing, this is known as the American Dream. The American Dream consistRead MoreThe Appeal And Effect Of Fantasy Essay1121 Words   |  5 Pagesand dramatists such as John Braine, Alan Sillitoe and Stan Barstow who were referenced as angry young men’. Notably, the writers were mostly young, working class and male, who responded to the disillusionment created by the perceived failure of post war administrations. They considered the labour government had failed to deliver an egalitarian society and allowed the continuation of an entrenched class system. The term ‘angry young man’, was originally coined in 1956, following the opening of JohnRead MoreFast Food Nation : The Dark Side Of The All American Meal1135 Words   |  5 PagesThe â€Å"Drive Thru† Consumerism Of The 1950’s In Eric Schlosser’s 2001 piece, Fast Food Nation: The Dark Side of the All-American Meal, he examines the rise of the fast food industry in the 1950’s as it was associated with the rampant consumerism of the era and shows how this led to the fast food industry becoming one of the most unethical, manipulative, and greedy industries that ever existed. Schlosser shows how fast food corporations, through mass appealing advertising, were able to manipulateRead MoreThe Absolute Value Of America1555 Words   |  7 Pagesbuying of miscellaneous objects on the internet, are just a few contributors to the most powerful â€Å"ism† that powers America. From the 1920s to the present day America has been driven by consumerism. Consumerism- in its simplest form- is defined as the buying and selling of products. When tracing the evolution of consumerism in America, one must explore many factors that led up to today’s consumerist culture; the economic ups and downs of t he 1920s through the 1950s, the anti-consumerist movement in theRead MoreFast Food Nation : The Dark Side Of The All American Meal1682 Words   |  7 PagesKaushal Brahmbhatt HIST 173 Recent US History December 10, 2015 The â€Å"Drive Thru† Consumerism Of The 1950’s In Eric Schlosser’s 2001 piece, Fast Food Nation: The Dark Side of the All-American Meal, he examines the rise of the fast food industry in the 1950’s as it was associated with the rampant consumerism of the era and shows how this led to the fast food industry becoming one of the most unethical, manipulative, and greedy industries that ever existed. Schlosser shows how fast food corporationsRead MoreThe Rise Of Consumerism During World War II1020 Words   |  5 Pagesservicemen returning home after World War II, the United States was filled with an energy that had long been repressed by an economic depression in the 1930’s. By the 1950’s, Americans were ready to move on from the war and start families. Thousands of jobs were created to accommodate all those retuning home, which caused the United States’ economy to flourish. Since more people were working and receiving higher wages, they were eager to start spending. Because of this, consumerism s kyrocketed in the 1950’sRead MoreFordism, Post-Fordism and the Flexible System of Production1199 Words   |  5 Pages------------------------------------------------- Top of Form Bottom of Form Other Free Encyclopedias  » Science Encyclopedia  » Science amp; Philosophy: Condensation to Cosh  » Consumerism - Consumerism And Mass Production, Consumerism And Post-fordism, Soap, The Politics Of Consumerism Consumerism - Consumerism And Post-fordism soap particular class world fordist consumption market mass Ads by Google Mr Power Giant Controller Saves 50% of your GEYSER costs! Pays for itself within months. www.mrpower.co.za Read MoreBooming Effect : The Baby Boomer Generation Essay1358 Words   |  6 Pages Booming Effect: The Baby Boomer Generation The ending of World War II led to one of the most influential generations today. Young males upon returning to the United States, Canada, and Australia following tours of duty overseas during World War II began families. This brought about a significant number of new children into the world. â€Å"In 1946, the first year of the Baby Boom, new births in the U.S. skyrocketed to 3.47 million births† (Rosenburg)Read MoreThe World s Strongest Military Power1580 Words   |  7 PagesBy the end of World War II, it was globally evident that the United States was the world’s strongest military power. During the 1950s, the United States experienced a period of glaring economic growth, with an increase in manufacturing and nation-wide consumerism. The benefits of this prosperity — television sets, new cars, new homes (suburbanization), and other consumer goods — were more prominent than ever before. The 50s were also an era of great conflict. (ex. racial discrimi nation and the earlyRead MoreA Social Examination On The Cold War969 Words   |  4 PagesBrittany O’Neill May Paper Elaine Tyler takes a social examination on the war against communism in the book, Homeward Bound: American Families in the Cold War Era. May portrays the idea that the nuclear family structure was a way to amplify resistance against communism. The exterior threat of communism during the postwar and the Cold War era caused for interrelationships within marriages to become a longer and more stable environment. Compared to the previous book we read as a class, May takes

Why School Curriculum Should Be A Multicultural Essay

Response to Prompt #1 From this week s reading of Teaching to Change the World, we learned of the demographic shift in public education. I believe this change is happening for the better. Like many institutions in our society, we must change with the times to meet the needs of the people we serve. School curriculum should be something that is always evolving, for the simple fact that students who our public schools are serving learn differently and at times, come with a different set of cultural norms and experiences with them that may impact the way they learn. My K-12 school experience was in an urban environment that was predominantly African American. In my first three years of teaching my school demographic has been similar to my own school experience. I believe curriculum should be multicultural in focus. For example, in my first teaching position at a charter school on the North side of Columbu, OH, my school s population had an overwhelming number of students from western and eastern Africa. The school represented the demographic shift that was occurring on the northside of Columbus. First year teaching is already an overwhelming experience, but add on not having any curriculum guidance and having to make everything from scratch with limited resources made me think outside of the box. I was teaching world history that year, my major focus of that year was making connections on how european colonialism and exploitation of groups around the world have and stillShow MoreRelatedStudents Are Not Entering The Classroom With The Knowledge And Understanding Of Multicultural Education762 Words   |  4 Pagesclassroom with the knowledge and understanding of the importance of multicultural literature. The students are suffering due to the lack of diverse reading literature incorporated into units of reading study. There is a need for staff developments and in-services to help educate teachers on ways to provide multicultural literacy awareness in primary schools. Objectives †¢ To heightening the awareness of the importance of multicultural literature in early grades. †¢ To prepare educators for diversityRead MoreQuestions and Answers on Leadership1168 Words   |  5 Pagesthings I gained from reading the chapters is that diversity has been and continues to be an issue in the public school setting. What can I do as a leader to enhance diversity in my school and what can I do to make the climate more multicultural? Those are questions that must be addressed and answered. As Koppelman (2014) states, â€Å"The challenge confronting us today is how to become multicultural individuals. In the teaching profession, that question will be answered by white-middle class individuals-Read MoreAfrican American History And Education Of All Perspectives Essay1522 Words   |  7 PagesStage 4: Structural Reform occurs when a school can provide new materials, and perspectives, seamlessly with the knowledge to provide new levels of understanding from a more complete and accurate curriculum. According to Gorski, Stage 4 is where a teacher dedicates her- or himself to continuously expanding her or his knowledge base through the exploration of various sources from various perspectives, and sharing that knowledge with her or his students. Students learn to view events, concepts, andRead MoreMulticultural Education int the United States1665 Words   |  7 Pagesdiversity. This influx has prompted school administrators to recognize the need to incorporate multicultural programs into their school environment including classroom settings, school wide activities, and curriculum as it becomes more evident that the benefits of teaching cultural diversity within the school setting will positively influence our communities, and ultimately the entire nation’s future. The purpose of this paper is to share the pros and cons of multicultural education in the classroom. AdditionallyRead MoreMulticultural Education : A Multicultural Classroom960 Words   |  4 PagesAn additional aspect to a perfect education system would be the use of mult icultural education in schools. Multicultural education creates a comfortable environment for students of all races and ethnicities to learn in by combining a variety of ideals about teaching. According to Geneva Gay, the creator of multicultural education, one of these ideals is understanding the cultural characteristics and cultural contributions of different ethnic groups, such as the values of different ethnic groups,Read MoreReasons For Asian High School Students782 Words   |  4 Pagesreasons why Asian high school students are not applying to colleges, this paper will discuss two reasons. First being, some Asian students are not able to pay for tuition. According to Gildersleeve (as cited by Hellen, 2002), for the past 20 years it has been more difficult for lower-income students to afford for college through merit-based financial aid in comparison to students who comes from middle to higher-income families. The second reason is because of their ethnicity. Which is why it is vitalRead MoreQuestions On Multicultural Education : The Material Presented Goes Along With Our Weekly Reading Assignments998 Words   |  4 Pagestopics into teaching and in my curriculum. I will be highlighting the presentations of Sarah, Virginia, Jessica and Ericka. Sarah Sponsel’s topic was multicultural education. Caleb Rosado shares that a school can be multicultural based on whether or not it uses the Five Ps. The five Ps are perspectives, policies, programs, personnel, and practices. The schools must also implement the four imperatives. The imperatives are: 1. Reflect the heterogeneity of the school; 2. Are sensitive to the needs ofRead MoreMulticultural Curriculum For A Multicultural Classroom1425 Words   |  6 PagesWhen I first started this class I was aware of multicultural curriculum but I was not aware of how important it was in a classroom setting. As the weeks have gone by in this class, I have learned that multicultural curriculum is important because it s a way for teachers to include all children from diverse backgrounds. As we ve have progressed in the study of multicultural curriculum we have learned to address important topics such as biases, social justice, stereotypes, the development of identityRead MoreEssay about Improving Education through Cultural Diversity1087 Words   |  5 Pagescultural diversity is important as it was many centuries ago. According to dictionary, cultural diversity is the coexistence of different culture, ethnic, race, gender in one specific unit. In order, for America to be successful, our world must be a multicultural world. This existence starts within our learning facilities where our students and children are educated. This thesis is â€Å"changing the way America, sees education through cultural diversity, has been co existing in many countries across the worldRead MoreEssay about Dr. James Banks on Multicultural Education1050 Words   |  5 Pagessociety. Dr. James A. Banks defines the meaning of multicultural education and its potential impact on society when it is truly integrated into American classrooms. In his lecture, Democracy, Diversity and Social Justice: Education in a Global Age, Banks (2006) defines the five dimensions of multicultural educ ation that serve as a guide to school reform when trying to implement multicultural education (Banks 2010). The goal of multicultural education is to encourage students to value their own

Essay Is Learning a Science - 801 Words

| Is Learning a Science? | Michelle L Yernest | Georgia Northwestern Technical College | | | | Science is the knowledge gained by careful observation, by deduction of the laws that govern changes and conditions, and by testing these deductions by doing experiments and then refining these experiments and testing them again. There are a couple of diverse learning methods. These methods are classical conditioning and operant conditioning. Precisely, what is conditioning? Conditioning is the process of changing behavior in such a way that an action formerly associated with a particular stimulus becomes associated with a new and unrelated stimulus. Both of these learning methods are basic forms of learning, which leads me†¦show more content†¦It appeared that the random actions of the cat leading to the opening of the door had become strengthened by its positive consequences. The reactions dwindled and probably will stop totally when the food reward is not disposed. The term for this is extinction (Conditioning-Classical and Operant Conditioning n.d. p. 1709). In this expe riment the reaction being conditioned, opening the latch is the operant because it functions on the environment. The reward or any consequence that makes a behavior stronger is the reinforcer of conditioning. Reinforcement is the procedure of withdrawing or offering negative or positive reinforcers to sustain or increase a response. It can occur after every response, this is called continuous. If it occurs only after some responses it is called intermittent (Operant Conditioning October 22, 2010). B. F. Skinner was a Harvard Psychologist who did the majority of the research on operant conditioning, although he did not â€Å"discover† it. He designed the operant box which is sometimes called the Skinner box. Skinner used rats in his research. Inside of the Skinner box there was a lever and a cup that stuck out of the wall. When the lever was pressed down, it would release a food pellet into the cup. Once the rat recognized that food was being released by pressing the lever, th e rate in which the lever gets pressed is increased. Learning has beenShow MoreRelatedScience As An Environment For Learning959 Words   |  4 PagesEven though science can be very mind boggling it is often at times misunderstood by many, it draws together model experimental developments and concepts in conjunction with matter, gadgets, and other devices because scientists contributes different conveniences for the different communities to develop and expand their individual awareness. It helps one to better understand the invention of recreating mass and matter. As an attempt to look at the continuing debate of the true role science plays inRead MoreThe Learning Area Of Science2183 Words   |  9 PagesThis is a learning story I published, involving two toddlers, one aged 2 years and three months and the other one year and eight months. It is in an outdoor environment, consent to use this story is added to my appendix. The following learning story shows the learning area of sc ience. Collins today I noticed you swinging on the swing and going really high. Your friend on the swing next to you was getting frustrated and upset as they could not get the swing to swing. Collins you slowed your swingRead MoreLearning And Science Inquiry Skills921 Words   |  4 Pages Learning dispositions and Science Inquiry Skills: Carr (2008, Para.2) says: â€Å"It is not about the blocks or the dough. It is about the activity being the vehicle for the acquisition of the disposition to learn.† Dispositions are voluntary and frequent habits of doing and thinking. They are environmentally sensitive as they could be acquired and supported by the interactive experiences in an environment with adults, peers and the nature around them (Bertram Pascal 2002; Aitken, Hunt, RoyRead MoreBilingual Course, Essentials Of The Learning Sciences Essay1082 Words   |  5 PagesWe had the 13th Class yesterday for the bilingual course â€Å"Essentials of the Learning Sciences†, and the topic of this week was about â€Å"Problem Based Learning (PBL)†. We were the discussion group, and it was the last time for our group to play the teacher roles in this bilingual course. Before the class, I made a tremendous effort in preparation for the class; reading the Chapter 15 of the CHLSv2, and searching for some definitions of some vocabularies. As results of the lack knowledge in this chapterRead MoreThe Paradox Of Science : A Contrivance For Childrens Learning2084 Words   |  9 PagesPart a: The paradox of science: a contrivance for children’s learning A cornerstone of the sciencing discourses evident in early childhood education and care settings revolves significantly around the children, educators and families of the service. It is therefore essential to explore teaching methods utilised with children to harness optimum engagement of children’s interest in science. Within this paper teaching methods are explored and discussed with an emphasis on the sciencing of sociologyRead MoreThe Effect Of Technology On Students Enthusiasm For Learning Science821 Words   |  4 Pageschanging and nowadays technology is being used in the classroom† (Daniel, 2011). Students learning with technology. â€Å"The effect of technology on students’ enthusiasm for learning science (both at school and away from school) was investigated† (Hollis, 1995). It will foster a great learning experience for the students. â€Å"Enthusiasm for learning science can be defined as the students’ eagerness to participate in s cience activities in the classroom, as well as away from school† (Hollis, 1995). Using technologyRead MoreImpact Of Technology On Student s Enthusiasm For Learning Science904 Words   |  4 PagesThe purpose of this study is to see how the integration of technology would impact the student’s enthusiasm for learning science in Mr. Hollis’s Science classroom. The integration of technology will involve teaching and learning that will foster a learning experience that will help students develop the knowledge and the skills to promote technology literacy. Per Hollis, he was motivated and had interests in integrating technology equipment and software in his curriculum using multimedia computerRead MoreEffects Of Technology On Young Learners Enthusiasm For Learning Science1061 Words   |  5 PagesHollis’s 1995 study focused on the effects of technology on young learner’s enthusiasm for learning science inside and outside of the classroom. This area of focus studied how implementing technology to teach science concepts impacted student’s motivation for learning science in the classroom. The teacher researcher’s study involves both teaching and learning as it focused on properly training teachers how to use and implement technological tools and software. Once teachers knew how to efficientlyRead MoreObservational Case Study: Student Learning in the Social Sciences2414 Words   |  10 Pagesï » ¿Observational Case Study of Student Learning in the Social Sciences Overview The case study reported herein this work in writing relates a case study, which is an observational study of humanities teaching and student learning in the social sciences. This study observes classroom instruction to identify issues with the teaching of humanities, student engagement and learning, what teachers and students do in the classroom context, and finally makes recommendations and identifies possible solutionsRead MoreLearning About Space During Our Science Block Essay1952 Words   |  8 PagesConstructivism A constructivist learning activity that I would develop for this class or a class similar to it for a third grade classroom would be inquiry based learning. In this third grade classroom the students would be learning about space. Inquiry based learning would be great for a third grade classroom as they are able to do research through books, the internet and by asking many questions. First, I would start out with introducing the topic of space during our science block. I would possibly start

Karl Marx (1958 words) Essay Example For Students

Karl Marx (1958 words) Essay Karl MarxannonKarl Marx was the greatest thinker and philosopher of his time. His viewson life and the social structure of his time revolutionized the way inwhich people think. He created an opportunity for the lower class to riseabove the aristocrats and failed due to the creation of the middle class. Despite this failure, he was still a great political leader and set thebasis of Communism in Russia. His life contributed to the way people thinktoday, and because of him people are more open to suggestion and arequicker to create ideas on political issues. Karl Heinrich Marx was born May 5th, 1818 in Trier. Although hehad three other siblings, all sisters, he was the favorite child to hisfather, Heinrich. His mother, a Dutch Jewess named Henrietta Pressburg,had no interest in Karls intellectual side during his life. His fatherwas a Jewish lawyer, and before his death in 1838, converted his family toChristianity to preserve his job with the Prussian state. When Heinrichsmother died, he no longer felt he had an obligation to his religion, thushelping him in the decision in turning to Christianity. Karls childhood was a happy and care-free one. His parents had agood relationship and it help set Karl in the right direction. His‘splendid natural gifts awakened in his father the hope that they wouldone day be used in the service of humanity, whilst his mother declared himto be a child of fortune in whose hands everything would go well. (Thestory of his life, Mehring, page 2)In High school Karl stood out among the crowd. When asked to writea report on How to choose a profession he took a different approach. Hetook the angle in which most interested him, by saying that there was noway to choose a profession, but because of circumstances one is placed inan occupation. A person with a aristocratic background is more likely tohave a higher role in society as apposed to someone from a much poorerbackground. While at Bonn at the age of eighteen he got engaged to Jenny vonWestphalen, daughter of the upperclassmen Ludwig von Westphalen. She wasthe childhood friend of Marxs oldest sister, Sophie. The engagement was asecret one, meaning they got engaged without asking permission of Jennysparents. Heinrich Marx was uneasy about this but before long the consentwas given. Karls school life other than his marks is unknown. He never spokeof his friends as a youth, and no one has ever came to speak of himthrough his life. He left high school in August of 1835 to go on to theUniversity of Bonn in the fall of the same year to study law. His fatherwanted him to be a lawyer much like himself but when Karls recklessuniversity life was getting in the way after a year Heinrich transferredhim to Berlin. Also, he did not go to most lectures, and showed littleinterest in what was to be learned. Karls reckless ways were nottolerated at Berlin, a more conservative college without the mischievousways of the other universities. While at Berlin, Marx became part of the group known as the YongHegelians. The group was organized in part due to the philosophy teacherHegel that taught from 1818 to his death. The teachings of Hegel shapedthe way the school thought towards most things. Those who studied Hegeland his ideals were known as the Young Hegelians. Hegel spoke of thedevelopment and evolution of the mind and of ideas. Although Karl wasyounger than most in the group, he was recognized for his intellectualability and became the focus of the group. While at Berlin He came tobelieve that all the various sciences and philosophies were part of oneoverarching, which, when completed, which would give a true and totalpicture of the universe and man. (Communist Manifesto, Marx (Francis B. Randal), page 15) Marx was an atheist, and believed that science andphilosophy would prove everything. Thus he had no belief in a god of anytype. Marx believed that Hegel must have been an atheist as well becauseof his strong belief in the mind. The Effects Of Video Games On The Heart EssayIn the second part Marx discusses the importance of Communism, andif private property is abolished, class distinctions will be as well. Thesecond part also stresses the importance of the necessity of theproletariat and bourgeoisie being common and the level of class being thesame. The third part critiques other social ideas of the modern day. Thefinal and fourth part discussed the differences between his politicalissues as apposed to those of the other oppositonal parties. This partends in bold capital letters WORKINGMEN OF ALL COUNTRIES, UNITE!The days of November 1850 fall almost exactly in the middle ofMarxs life and they represent, not only externally, an important turningpoint in his lifes work. Marx himself was keenly aware of this and Englesperhaps even more so. (The Story of his life, Mehring, page 208) Living inpolitical exile his life changed. His ideas were no longer followed likethey once were. His isolation from the general public provided a new lightin his life. Then, in 1855, his only son died. His son showed much potential,and was the life of the family. When he died, Jenny became very sick withanxiety, and Marx himself became very depressed. He wrote to Engles Thehouse seems empty and deserted since the boy died. He was its life andsoul. It is impossible to describe how much we miss him all of the time. Ihave suffered all sorts of misfortunes but now I know what real misfortuneis. (The Story of his Life, Mehring, page 247)After the Communist League disbanded in 1852 Marx tried to createanother organization much like it. Then, in 1862 the First Internationalwas established in London. Marx was the leader. He made the inauguralspeech and governed the work of the governing body of the International. When the International declined, Marx recommended moving it to the UnitedStates. The ending of the International in 1878 took much out of Marx, andmade him withdraw from his work; much like the ending of the CommunistLeague had done. This time, it was for good. The last ten years of his life is known as a slow death. This isbecause the last eight years many medical problems affected his life. Inthe autumn of 1873 he was inflected by apoplexy which effected his brainwhich made him incapable of work and any desire to write. After weeks oftreatment in Manchester, he recovered fully. He controlled the demise ofhis health. Instead of relaxing in his old age he went back to work on hisown studies. His late nights and early mornings decreased his health inthe last few years of his life. In January of 1883, after the death of hisdaughter Jenny, he suffered from Bronchitis and made it almost impossibleto swallow. The next month a tumor developed in his lung and soonmanifested into his death on March 14, 1883. Although Marxs influence was not great during his life, after hisdeath his works grew with the strength of the working class. His ideas andtheories became known as Marxism, and has been used to shape the ideas ofmost European and Asian countries. The strength of the Proletariat hasbeen due to the work of Marx. His ideals formed government known asCommunism. Although he was never a rich man, his knowledge has been richin importance for the struggle of the working class. Himelfarb, Alexander and C. James Richardson. Sociology for Canadians:Images of society. Toronto: McGraw-Hill Ryderson Limited, 1991Mehring, F, Karl Marx, The story of his life, London: Butler and Tannerltd., 1936Marx, K, The Communist Manifesto, Germany: J. E. Burghard, 1848Karl Marx. Microsoft Encarta 96 Encyclopedia. Cd-Rom. Microsoft Corp.,1993-1995Vesaey, G. and P. Foulkes. Collins dictionary of Philosophy. London:British Library Cataloguing in Publication Data, 1990

Human Inheritance free essay sample

Ethical dilemmas are constantly confronting healthcare professionals, which is difficult to deal with as there is no correct solution. These are also known as moral dilemmas as they are situations where there is more than two choices to make and none of the choices is certain to work and can cause complications. An example of this would be ‘You are a patient and are too sick to speak for yourself. You are concerned about who will make medical decisions on your behalf, and whether your wishes will be followed. You wonder, What if they disagree about what I would want, or what would be best for me? ’. Another example of this would be with the economic downturn that you may not be able to afford the funds for food and need to feed your family but the only way in doing this is to steal or let your family starve. These dilemmas are impossible because each person thinks differently and has a different feeling towards it. We will write a custom essay sample on Human Inheritance or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page There are ethical dilemmas surrounding IVF and infertility. Infertility is a genetic problem that affects women; it is not the woman’s fault. With IVF the NHS only gives each woman one free cycle and after that she has to fund it herself. An ethical dilemma with IVF is the possible wrong that is done to the infertile couple or the expected child by the physician. The success of IVF depends on the number of embryos transferred to the woman’s uterus. Because the chance of survival of an embryo in IVF is small the more transfers made the greater the chance of the woman becoming pregnant, it also increases the risk of multiple pregnancies. IVF is not allowed by the Catholic Church because it separates the unitive and the procreative aspects of marriage. To separate the unitive and the procreative aspects of marriage is a mortal sin. In addition the sperm donor commits a mortal sin in order to harvest the sperm which is needed for IVF. Although one human life may be created through the IVF technique, many surplus foetuses, (unborn babies), are destroyed through this process. Other surplus unborn babies are left frozen in the laboratories where they were manufactured as though they were not human beings, but simply consumer goods. They were not created in love through the marriage act as God intends. Multiple births also create danger to the health and well-being of the child. Premature birth and low weight when born are also issues with this, also studies that have been undertaken spina bifida is at a higher risk with children made from IVF. Also the hormones that are taken by the female in order to become pregnant are always at risk of having problems or abnormalities to the unborn child. Aminiocentesis is another ethical dilemma, during the process if abnormalities are found the mother is offered the chance to terminate her pregnancy. The ethical issues surrounding amniocentesis are seen as centring on 4 focal points. First is the policy of the diagnostic treatment centre. Here, 2 questions arise: Is the client involved in a high-risk pregnancy? And, if a positive diagnosis is made, will the parents’ consent to an abortion? Second is the role of the genetic counsellor, which is seen as supportive rather than leading. He should assist the prospective parents in reaching a decision to undergo amniocentesis and possible abortion that is mutually acceptable. The prospective parents, the third focal point, may face the question of deciding what is normal. The clients must also realize the terrible strains that are put on a marriage into which a severely defective child has been born. The fourth focal point is public policy. While amniocentesis may appear to threaten some values held important in our society, the author regards the procedure as an interim solution on the road to an understanding of and ability to treat genetic defects. Contraception is another ethical dilemma as birth control operates before pregnancy begins, and until the sperm fertilises the egg there is nothing that is going to suffer loss and so the issue is very different from the case of abortion. And since the egg and sperm would cease to exist whether fertilisation takes place or not, they cant be said to suffer loss, either. Non-religious arguments about birth control are therefore concerned only with the rights of the parents and with the consequences for those parents and for society in general. The issue of possibly killing a person, and of the rights of the mother versus the rights of the foetus, which dominate the topic of abortion, do not arise. Some people think it’s wrong as it is wrong to interfere with the natural order of the universe. People in certain religions also see it as wrong because of the fact that it is like abortion as some birth control techniques can operate by preventing the implantation and development of a fertilized egg. Those opposed to such methods say that this amounts to an abortion, and that if abortion is wrong then those forms of contraception must also be wrong. http://brendakaren. wordpress. com/2009/04/15/some-moral-and-ethical-issues-concerning-ivf-techniques/ http://www. ncbi. nlm. nih. gov/pubmed/4418247 Contraception!!! http://www. bbc. co. uk/ethics/contraception/contraception_abortion. shtml

King Lear Essays - King Lear, Cordelia, Goneril, Fool, Edmund, Regan

King Lear Essays - King Lear, Cordelia, Goneril, Fool, Edmund, Regan King Lear King Lear Historians en masse have determined that Shakespeare was most definitely not the first one to come up with the general plot lines contained in King Lear. Though the play revolves mainly around the conflict between the King and his daughters, there is a definite and distinct sub-plot dealing with the plight and tragedy of Gloucester as well. The play (both stories really) has origins in many different sixteenth century works, with nearly all the pertinent facts such as the name of the King, the three daughters, their husbands, the answers of the three daughters when Lear asks them to profess their love, Cordelia's ensuing disgrace, and the cruelty of the two dukes and duchesses to the King contained in Raphael Holinshed's Chronicles. (Chapters five and seven of the Second Book of the History of England, second ed., 1587) Shakespeare is also believed to have borrowed, significantly less however, from a play that was entered in the Stationer's Register, 14! May 1594, called, The moste famous Chronicle historye of Leire kinge of England and his Three Daughters. This piece was considered to be "quite un-Shakespearian" and untragical, and was entered subsequently on the Stationer's Register as The Tragecall historie of Kinge Leir and his Three Daughters, as it was latelie acted. Much of Shakespeare's account of the Gloucester story was borrowed from Sir Philip Sydney's Arcadia, 1590. In terms of the Gloucester-Edmund-Edgar plot, we can find many similarities in the second book of Arcadia, chapter ten, in a narrative called, The pitifull state, and story of the Paphlagonian unkind king, and his kind son, first related by the son, then by the blind father. The main difference here, of course, is that Shakespeare has intertwined this plot with the plight of Lear and his three daughters. There are many differences between these texts and the Shakespearian version of King Lear. None of these earlier works had the signature character of the Fool, and Shakespeare creatively transformed what was known earlier as a, "melodramatic story with a 'happy ending'," into a biting and, above all else, sad story of the relationship between parents and their children. One of the main themes that Shakespeare chooses to focus on in King Lear is the dysfunctional nature of not only the royal family and Gloucester, but the heartache and emotional strain that goes along with being a parent and having to make a decision that will divide your children. This play focuses on not only the after effects of this decision, but the way in which it affects the King, his children and his subjects as well. A strong case can be made for King Lear as Shakespeare's most tragic effort of his career. The fact that nearly the entire cast of this play either is murdered or dies with little to no redemption makes the strongest case for this. In nearly every other Shakespearian work, save perhaps Othello, at least some of the characters enjoy a bit of redemption or salvation with the resolution of the conflict. King Lear's characters are privy to neither of these. The bitterness, sadness, and reality of the human psyche that is contained throughout this work demonstrate its tragic nature best, however. The tie emotionally and physically between a father and a daughter (or son, in relation to the Gloucester/Edmund/Edgar plot) is something entirely different than husband-wife or boyfriend-girlfriend in many of Shakespeares other plays. In the very beginning of the play, when Lear is foolishly dividing up his kingdom between his three daughters, and after he has asked Cordelia's two older sisters what they "think" of him, he turns to her and asks the same question. Her reply shows the true nature of her character, as she says, "Unhappy that I am, I cannot heave my heart into my mouth. I love Your Majesty according to my bond, no more, nor less." (1.1, ll. 91-93) His words could almost be considered threatening by declaring that her unwillingness to express her love in words might, "mar her fortunes." We are privy to definitive foreshadowing with Cordelia's reply of, "Good my lord, you have begot me, bred me, loved me. I return those duties back to you as are right

ART Essay Example | Topics and Well Written Essays - 250 words

ART - Essay Example Three old ladies were depicted in the oil painting among them two looked really dynamic; one painter, a model, and a portrait which being drawn. The model and the painter looked very real whereas, the portrait of the lady indeed had all features of a comparatively beautiful painting. The notable thing was that the emotional vigor reflected on the faces of the model and the lady in the painting was really difficult for me to understand. The picture also contained the essential furniture and fittings of the interior of a house. The painter in the painting seemed calm and contented as if she was reflecting on the beauty of her work. In fact, I did not like any other paintings because I could not comprehend what many of them meant. The Core New art space is a place I would suggest everyone to visit. Numerous pictures are well arranged there with detailed information on each one. We can purchase them at affordable prices and can get corporate or supporting membership to support the progra m of Core. The gallery is an excellent place for a person to learn many things about art. Each picture displayed holds every detail including the painter, materials used, date etc. As mentioned above, I chose this picture for its unusual appeal and the idea presented in it.

Thesis Final Paper Example | Topics and Well Written Essays - 1000 words

Final Paper - Thesis Example Special needs students should likewise mingle with other students and partake in peer and teacher socializations. An effective technique for the integration of special needs students entails simply a cue not to treat them as special persons with disabilities. In this process, students with disabilities achieve a sense of belonging and a feeling of acceptance (Cooper et al, 2002). This paper will discuss how special needs students learn through the help of inclusion and accommodation in the regular classroom settings. This paper will also prove that teachers play an important role in educating the special needs students. Academic institutions, as well as educators, are expected to make adjustments for the diversities of special needs students by modifying features in the school setting that may be unfavorable to the students advancement. Accommodation refers to modification of the school facilities, programs, and training in relation to education of the disabled students. In court cases, accommodation has been used to refer to amendments in the special needs students’ education. Likewise, accommodation refers to advances wherein several components of the entire learning environment of the students are modified for more education encouragement. The educators emphasize on amending the educational atmosphere or the learning necessities to enable these students to be educated regardless of their limitations or deficits (Price et al, 1998). Accommodation entails the utilization of customized training practices, more bendable administrative methods, adaptable educational conditions, or any classroom activity that focuses on the use of more integral abilities or that offers revised educational processes (Price et al, 1998). The majority of students, particularly the ones with learning disabilities, can profit from study skills training. The

History of Policing Essay Example for Free

History of Policing Essay The function of policing has played a considerable role in American history. The policing occupation has worked toward protecting citizens’ rights and helping America to become the free nation it is today. The United States of America is built from the U.S. Constitution and its Bill of Rights, from this document we gather the rules of policing and make sure that every Americans rights are met. The evolutions of policing practices that officers have learned have changed American history for the better. As new problems in society arise, police must change and adapt to protect and serve the public. Early American policing strategies were based off of a similar British model. Law enforcement was not well organized or structured until 1200 A.D., after that time offenders were being pursued by an organized posse. In those times offenders were caught be the organized posse and were usually tortured and faced public execution. Rarely were the criminals or the accused of this time given the right to fair trail, and were not considered innocent until proven guilty like the laws we have today state, it was the other way around, where citizens that were accused of crimes were guilty before any evidence or testimony were ever provided. One of the earliest forms of policing came about in English cities and towns and were called night watches.(Schmalleger, 2009). The primary purpose of a night watch was to watch out for fires and thievery. There was also a day watch which basically was the same job as night watch, but in the day. Eventually this form of policing led to a written law being proposed in 1285 called the Statute of Winchester. This law created a watch and ward system that gave early watches a systems and structure to form themselves around. This law gave early English towns the policing practices and guidelines needed to produce a stable society, free of criminal activities, allowing the towns the type of policing that they needed to thrive. The Statute of Winchester law consisted of four main points, things that were specific to the watch of that town, the mandating of age eligible men to serve, institutionalizing the use of the hue and cry, and for answering the call of duty. Prior structuring and laws such as the Statue of Winchester propelled law enforcement into its future when prime minister of England Sir Robert Peel formed the world’s first modern police force. Peel’s model of new police became the model for police all across the globe. He formed the police with more of a military outline; giving uniforms for better origination and structure. Early American leaders followed the day and night watch approach, later American leaders followed Sir Robert Peel method, which is what American policing still utilize today. New technological advances and social reform for policing were brought about in the twentieth century. The invention of automobiles, telephones, and radios were developments of the twentieth century, which have helped police officers with their communications, speed of investigations and coordination of efforts. Automobiles allowed police a quick responds and allowed them to serve greater areas. Radios and telephones allowed for the communication and coordination of efforts between police officers. Teddy Roosevelt contributed his part of advancement of the policing system by organizing the FBI and helping to promote the first call box system, which is closely associated with the 911 system used today. The political era (1840s-1930) was an era that policing organizations were more concerned with the interest of powerful politicians rather than the rights and laws of the people. The next eras of policing came from 1930s-1970s where police became increasingly concerned with solving of more traditional style of crimes. The time from the 1970s to the end of the twentieth century is considered to be the third period of policing and the most contemporary of policing practices, acting on policing of each community. Finally present day policing is or the modern era has given way to that of homeland security, which grew after the September 11, 2001 terrorist attack on the United States. All levels of law enforcement agencies, local; state; and federal have devoted more time and effort toward the protection of our nation and homeland security, in an effort to thwart any future attacks. Local, state, and federal law enforcement are tasked with the enforcement of the laws. Federal law enforcement agencies are government agencies whose primary function is the protection and enforcement of federal laws. State law enforcement agencies were created for specified set of needs. State law enforcement was built from one of two models. The first model combines major criminal investigations with the patrolling of state highways. The second model is the culmination of two functions which consist of traffic enforcement and other laws that need upheld. Today’s duties of the modern state policing are to assist local law enforcement in criminal investigations, operate identification bureaus, maintain criminal records repository, patrol states highways, and provide training for municipals and county officers. Today they have also stepped up to a role of homeland security, for identifying precursors to acts of terrorism. The third level of law enforcement is that of the local policing authorities, i.e. county sheriffs or town marshals. City and county agencies are both part of the local level, mayors or city council appoint the officers to their offices and their jurisdictions are limited only to the boundary of their communities. Sheriffs are responsible for law enforcement in unincorporated areas and for the operation of the county jail. Local police play their role in the new era of homeland security as well by ascertaining the changes in the community that are out of place or unusual. Local police help do this task by interacting with the public, and constantly patrolling neighborhoods and other parts of their community. The increases of population, crime, and the advancement of technologies have challenged our policing forces but it has also allowed it to make necessary changes and advancements in law enforcement evolve with the times, with each generation of Americans we continue to increase our knowledge, experience and technological advances. The structure and technologies of the policing may change over a given time but the ideal of protecting and serving the community will not. With increasing threats on the rise, the police force must be capable and flexible. Making sure they have the most innovative technologies to oppose impending threats and overcome the future challenges that may lie ahead. Reference Schmalleger, F. (2009). Criminal Justice Today: an Introductory Text for the 21st Century (10th ed.). New York, NY: Prentice-Hall. 2007. DEPARTMENT OF HOMELAND SECURITY: Progress Report on Implementation of Mission and Management Functions. GAO Reports i. MasterFILE Premier, EBSCOhost (accessed June 12, 2011). Dodsworth, F. M. (2008). The Idea of Police in Eighteenth-Century England: Discipline, Reformation, Superintendence, c. 1780-1800. Journal of the History of Ideas, 69(4), 583-604. Retrieved from EBSCOhost. Scott, J. E. (2010). Evolving Strategies: A Historical Examination of Changes in Principle, Authority and Function to Inform Policing in the Twenty-First Century. Police Journal, 83(2), 126-163. doi:10.1350/pojo.2010.83.2.490

The Effect of Technology on My Life :: Exploratory Essays Research Papers

The Effect of Technology on My Life I roll around on my bed, tossing and turning. The blare from my alarm clock deafens my right ear, and I quickly throw an arm over to it and slam on the snooze button. It is 6 o'clock in the morning, and already technology has affected my life. I fall to my feet and walk towards the showers. Another form of technology is about to take over my life. Well, at least for the next ten to 20 minutes. The alarm clock, running water, these are only two of the millions of examples of technology I will encounter today. I place technology into two, well, three basic categories: Informative, which helps us obtain and use information, Communicative, which includes language, signs, and the like, and that help us communicate with each other, and Useful, such as electricity and running water. These are the things that make our lives easier and help us get through the day to day. Also, there are those things that we are taught. I believe that learning, such as being racist, is a technology. Okay, so I have four categories, all right? Education and Learning is the fourth. I believe this because you are not born a racist, you're your environment and those you are around teach you to be that way. Also, it ties in to other forms of technology. Like, when you're a baby, you can't speak, so you have no way to communicate how you fell. But as you grow older, you learn language t hat you hear being spoken around you. You slowly pick up on the words and phrases used to express desires and thoughts. This is learning and teaching as technology. This "story" is about how my life is affected by technology, and so I got to thinking: What better way to explain this than to take you (the reader) on a journey through a typical day in my life? Here we go. I've already woken up, and taken my shower, and now I trudge back to my room to get ready for my classes. I turn on my lamp (technology), without which I couldn't put on my make-up.

The Higher Arithmetic – an Introduction to the Theory of Numbers

This page intentionally left blank Now into its eighth edition and with additional material on primality testing, written by J. H. Davenport, The Higher Arithmetic introduces concepts and theorems in a way that does not require the reader to have an in-depth knowledge of the theory of numbers but also touches upon matters of deep mathematical signi? cance. A companion website (www. cambridge. org/davenport) provides more details of the latest advances and sample code for important algorithms. Reviews of earlier editions: ‘. . . the well-known and charming introduction to number theory . . can be recommended both for independent study and as a reference text for a general mathematical audience. ’ European Maths Society Journal ‘Although this book is not written as a textbook but rather as a work for the general reader, it could certainly be used as a textbook for an undergraduate course in number theory and, in the reviewer’s opinion, is far superior for this purpose to any other book in English. ’ Bulletin of the American Mathematical Society THE HIGHER ARITHMETIC AN INTRODUCTION TO THE THEORY OF NUMBERS Eighth edition H. Davenport M. A. , SC. D. F. R. S. late Rouse Ball Professor of Mathematics in the University of Cambridge and Fellow of Trinity College Editing and additional material by James H. Davenport CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www. cambridge. org Information on this title: www. cambridge. org/9780521722360  © The estate of H. Davenport 2008 This publication is in copyright.Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008 ISBN-13 ISBN-13 978-0-511-45555-1 978-0-521-72236-0 eBook (EBL) paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. CONTENTS Introduction I Factorization and the Primes 1. 2. 3. 4. . 6. 7. 8. 9. 10. The laws of arithmetic Proof by induction Prime numbers The fundamental theorem of arithmetic Consequences of the fundamental theorem Euclid’s algorithm Another proof of the fundamental theorem A property of the H. C. F Factorizing a number The series of primes page viii 1 1 6 8 9 12 16 18 19 22 25 31 31 33 35 37 40 41 42 45 46 II Congruences 1. 2. 3. 4. 5. 6. 7. 8. 9. The congruence notation Linear congruences Fermat’s theorem Euler’s function ? (m) Wilson’s theorem Algebraic congruences Congruences to a prime modulus Congr uences in several unknowns Congruences covering all numbers v vi III Quadratic Residues 1. 2. 3. 4. . 6. Primitive roots Indices Quadratic residues Gauss’s lemma The law of reciprocity The distribution of the quadratic residues Contents 49 49 53 55 58 59 63 68 68 70 72 74 77 78 82 83 86 92 94 99 103 103 104 108 111 114 116 116 117 120 122 124 126 128 131 133 IV Continued Fractions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Introduction The general continued fraction Euler’s rule The convergents to a continued fraction The equation ax ? by = 1 In? nite continued fractions Diophantine approximation Quadratic irrationals Purely periodic continued fractions Lagrange’s theorem Pell’s equation A geometrical interpretation of continued fractionsV Sums of Squares 1. 2. 3. 4. 5. Numbers representable by two squares Primes of the form 4k + 1 Constructions for x and y Representation by four squares Representation by three squares VI Quadratic Forms 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction Equivalent forms The discriminant The representation of a number by a form Three examples The reduction of positive de? nite forms The reduced forms The number of representations The class-number Contents VII Some Diophantine Equations 1. Introduction 2. The equation x 2 + y 2 = z 2 3. The equation ax 2 + by 2 = z 2 4. Elliptic equations and curves 5.Elliptic equations modulo primes 6. Fermat’s Last Theorem 7. The equation x 3 + y 3 = z 3 + w 3 8. Further developments vii 137 137 138 140 145 151 154 157 159 165 165 168 173 179 185 188 194 199 200 209 222 225 235 237 VIII Computers and Number Theory 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction Testing for primality ‘Random’ number generators Pollard’s factoring methods Factoring and primality via elliptic curves Factoring large numbers The Dif? e–Hellman cryptographic method The RSA cryptographic method Primality testing revisited Exercises Hints Answers Bibliography IndexINTRODUCTION T he higher arithmetic, or the theory of numbers, is concerned with the properties of the natural numbers 1, 2, 3, . . . . These numbers must have exercised human curiosity from a very early period; and in all the records of ancient civilizations there is evidence of some preoccupation with arithmetic over and above the needs of everyday life. But as a systematic and independent science, the higher arithmetic is entirely a creation of modern times, and can be said to date from the discoveries of Fermat (1601–1665).A peculiarity of the higher arithmetic is the great dif? culty which has often been experienced in proving simple general theorems which had been suggested quite naturally by numerical evidence. ‘It is just this,’ said Gauss, ‘which gives the higher arithmetic that magical charm which has made it the favourite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics. ’ The theory of numbers is generally considered to be the ‘purest’ branch of pure mathematics.It certainly has very few direct applications to other sciences, but it has one feature in common with them, namely the inspiration which it derives from experiment, which takes the form of testing possible general theorems by numerical examples. Such experiment, though necessary in some form to progress in every part of mathematics, has played a greater part in the development of the theory of numbers than elsewhere; for in other branches of mathematics the evidence found in this way is too often fragmentary and misleading.As regards the present book, the author is well aware that it will not be read without effort by those who are not, in some sense at least, mathematicians. But the dif? culty is partly that of the subject itself. It cannot be evaded by using imperfect analogies, or by presenting the proofs in a way viii Introduction ix which may convey the main idea o f the argument, but is inaccurate in detail. The theory of numbers is by its nature the most exact of all the sciences, and demands exactness of thought and exposition from its devotees. The theorems and their proofs are often illustrated by numerical examples.These are generally of a very simple kind, and may be despised by those who enjoy numerical calculation. But the function of these examples is solely to illustrate the general theory, and the question of how arithmetical calculations can most effectively be carried out is beyond the scope of this book. The author is indebted to many friends, and most of all to Professor o Erd? s, Professor Mordell and Professor Rogers, for suggestions and corrections. He is also indebted to Captain Draim for permission to include an account of his algorithm.The material for the ? fth edition was prepared by Professor D. J. Lewis and Dr J. H. Davenport. The problems and answers are based on the suggestions of Professor R. K. Guy. Chapter VIII a nd the associated exercises were written for the sixth edition by Professor J. H. Davenport. For the seventh edition, he updated Chapter VII to mention Wiles’ proof of Fermat’s Last Theorem, and is grateful to Professor J. H. Silverman for his comments. For the eighth edition, many people contributed suggestions, notably Dr J. F. McKee and Dr G. K. Sankaran.Cambridge University Press kindly re-typeset the book for the eighth edition, which has allowed a few corrections and the preparation of an electronic complement: www. cambridge. org/davenport. References to further material in the electronic complement, when known at the time this book went to print, are marked thus:  ¦:0. I FACTORIZATION AND THE PRIMES 1. The laws of arithmetic The object of the higher arithmetic is to discover and to establish general propositions concerning the natural numbers 1, 2, 3, . . . of ordinary arithmetic. Examples of such propositions are the fundamental theorem (I. 4)? hat every nat ural number can be factorized into prime numbers in one and only one way, and Lagrange’s theorem (V. 4) that every natural number can be expressed as a sum of four or fewer perfect squares. We are not concerned with numerical calculations, except as illustrative examples, nor are we much concerned with numerical curiosities except where they are relevant to general propositions. We learn arithmetic experimentally in early childhood by playing with objects such as beads or marbles. We ? rst learn addition by combining two sets of objects into a single set, and later we learn multiplication, in the form of repeated addition.Gradually we learn how to calculate with numbers, and we become familiar with the laws of arithmetic: laws which probably carry more conviction to our minds than any other propositions in the whole range of human knowledge. The higher arithmetic is a deductive science, based on the laws of arithmetic which we all know, though we may never have seen them form ulated in general terms. They can be expressed as follows. ? References in this form are to chapters and sections of chapters of this book. 1 2 The Higher Arithmetic Addition.Any two natural numbers a and b have a sum, denoted by a + b, which is itself a natural number. The operation of addition satis? es the two laws: a+b =b+a (commutative law of addition), (associative law of addition), a + (b + c) = (a + b) + c the brackets in the last formula serving to indicate the way in which the operations are carried out. Multiplication. Any two natural numbers a and b have a product, denoted by a ? b or ab, which is itself a natural number. The operation of multiplication satis? es the two laws ab = ba a(bc) = (ab)c (commutative law of multiplication), (associative law of multiplication).There is also a law which involves operations both of addition and of multiplication: a(b + c) = ab + ac (the distributive law). Order. If a and b are any two natural numbers, then either a is equal to b o r a is less than b or b is less than a, and of these three possibilities exactly one must occur. The statement that a is less than b is expressed symbolically by a < b, and when this is the case we also say that b is greater than a, expressed by b > a. The fundamental law governing this notion of order is that if a b. We propose to investigate the common divisors of a and b.If a is divisible by b, then the common divisors of a and b consist simply of all divisors of b, and there is no more to be said. If a is not divisible by b, we can express a as a multiple of b together with a remainder less than b, that is a = qb + c, where c < b. (2) This is the process of ‘division with a remainder’, and expresses the fact that a, not being a multiple of b, must occur somewhere between two consecutive multiples of b. If a comes between qb and (q + 1)b, then a = qb + c, where 0 < c < b. It follows from the equation (2) that any common divisor of b and c is also a divisor of a.Moreo ver, any common divisor of a and b is also a divisor of c, since c = a ? qb. It follows that the common divisors of a and b, whatever they may be, are the same as the common divisors of b and c. The problem of ? nding the common divisors of a and b is reduced to the same problem for the numbers b and c, which are respectively less than a and b. The essence of the algorithm lies in the repetition of this argument. If b is divisible by c, the common divisors of b and c consist of all divisors of c. If not, we express b as b = r c + d, where d < c. (3)Again, the common divisors of b and c are the same as those of c and d. The process goes on until it terminates, and this can only happen when exact divisibility occurs, that is, when we come to a number in the sequence a, b, c, . . . , which is a divisor of the preceding number. It is plain that the process must terminate, for the decreasing sequence a, b, c, . . . of natural numbers cannot go on for ever. Factorization and the Primes 17 Let us suppose, for the sake of de? niteness, that the process terminates when we reach the number h, which is a divisor of the preceding number g.Then the last two equations of the series (2), (3), . . . are f = vg + h, g = wh. (4) (5) The common divisors of a and b are the same as those of b and c, or of c and d, and so on until we reach g and h. Since h divides g, the common divisors of g and h consist simply of all divisors of h. The number h can be identi? ed as being the last remainder in Euclid’s algorithm before exact divisibility occurs, i. e. the last non-zero remainder. We have therefore proved that the common divisors of two given natural numbers a and b consist of all divisors of a certain number h (the H. C. F. f a and b), and this number is the last non-zero remainder when Euclid’s algorithm is applied to a and b. As a numerical illustration, take the numbers 3132 and 7200 which were used in  §5. The algorithm runs as follows: 7200 = 2 ? 3132 + 936, 3 132 = 3 ? 936 + 324, 936 = 2 ? 324 + 288, 324 = 1 ? 288 + 36, 288 = 8 ? 36; and the H. C. F. is 36, the last remainder. It is often possible to shorten the working a little by using a negative remainder whenever this is numerically less than the corresponding positive remainder. In the above example, the last three steps could be replaced by 936 = 3 ? 324 ? 6, 324 = 9 ? 36. The reason why it is permissible to use negative remainders is that the argument that was applied to the equation (2) would be equally valid if that equation were a = qb ? c instead of a = qb + c. Two numbers are said to be relatively prime? if they have no common divisor except 1, or in other words if their H. C. F. is 1. This will be the case if and only if the last remainder, when Euclid’s algorithm is applied to the two numbers, is 1. ? This is, of course, the same de? nition as in  §5, but is repeated here because the present treatment is independent of that given previously. 8 7. Another proof of t he fundamental theorem The Higher Arithmetic We shall now use Euclid’s algorithm to give another proof of the fundamental theorem of arithmetic, independent of that given in  §4. We begin with a very simple remark, which may be thought to be too obvious to be worth making. Let a, b, n be any natural numbers. The highest common factor of na and nb is n times the highest common factor of a and b. However obvious this may seem, the reader will ? nd that it is not easy to give a proof of it without using either Euclid’s algorithm or the fundamental theorem of arithmetic.In fact the result follows at once from Euclid’s algorithm. We can suppose a > b. If we divide na by nb, the quotient is the same as before (namely q) and the remainder is nc instead of c. The equation (2) is replaced by na = q. nb + nc. The same applies to the later equations; they are all simply multiplied throughout by n. Finally, the last remainder, giving the H. C. F. of na and nb, is nh, wher e h is the H. C. F. of a and b. We apply this simple fact to prove the following theorem, often called Euclid’s theorem, since it occurs as Prop. 30 of Book VII.If a prime divides the product of two numbers, it must divide one of the numbers (or possibly both of them). Suppose the prime p divides the product na of two numbers, and does not divide a. The only factors of p are 1 and p, and therefore the only common factor of p and a is 1. Hence, by the theorem just proved, the H. C. F. of np and na is n. Now p divides np obviously, and divides na by hypothesis. Hence p is a common factor of np and na, and so is a factor of n, since we know that every common factor of two numbers is necessarily a factor of their H. C. F.We have therefore proved that if p divides na, and does not divide a, it must divide n; and this is Euclid’s theorem. The uniqueness of factorization into primes now follows. For suppose a number n has two factorizations, say n = pqr . . . = p q r . . . , where all the numbers p, q, r, . . . , p , q , r , . . . are primes. Since p divides the product p (q r . . . ) it must divide either p or q r . . . . If p divides p then p = p since both numbers are primes. If p divides q r . . . we repeat the argument, and ultimately reach the conclusion that p must equal one of the primes p , q , r , . . . We can cancel the common prime p from the two representations, and start again with one of those left, say q. Eventually it follows that all the primes on the left are the same as those on the right, and the two representations are the same. Factorization and the Primes 19 This is the alternative proof of the uniqueness of factorization into primes, which was referred to in  §4. It has the merit of resting on a general theory (that of Euclid’s algorithm) rather than on a special device such as that used in  §4. On the other hand, it is longer and less direct. 8. A property of the H. C.F From Euclid’s algorithm one can deduce a remarkable property of the H. C. F. , which is not at all apparent from the original construction for the H. C. F. by factorization into primes ( §5). The property is that the highest common factor h of two natural numbers a and b is representable as the difference between a multiple of a and a multiple of b, that is h = ax ? by where x and y are natural numbers. Since a and b are both multiples of h, any number of the form ax ? by is necessarily a multiple of h; and what the result asserts is that there are some values of x and y for which ax ? y is actually equal to h. Before giving the proof, it is convenient to note some properties of numbers representable as ax ? by. In the ? rst place, a number so representable can also be represented as by ? ax , where x and y are natural numbers. For the two expressions will be equal if a(x + x ) = b(y + y ); and this can be ensured by taking any number m and de? ning x and y by x + x = mb, y + y = ma. These numbers x and y will be natura l numbers provided m is suf? ciently large, so that mb > x and ma > y. If x and y are de? ned in this way, then ax ? by = by ? x . We say that a number is linearly dependent on a and b if it is representable as ax ? by. The result just proved shows that linear dependence on a and b is not affected by interchanging a and b. There are two further simple facts about linear dependence. If a number is linearly dependent on a and b, then so is any multiple of that number, for k(ax ? by) = a. kx ? b. ky. Also the sum of two numbers that are each linearly dependent on a and b is itself linearly dependent on a and b, since (ax1 ? by1 ) + (ax2 ? by2 ) = a(x1 + x2 ) ? b(y1 + y2 ). 20 The Higher ArithmeticThe same applies to the difference of two numbers: to see this, write the second number as by2 ? ax2 , in accordance with the earlier remark, before subtracting it. Then we get (ax1 ? by1 ) ? (by2 ? ax2 ) = a(x1 + x2 ) ? b(y1 + y2 ). So the property of linear dependence on a and b is preserved by addition and subtraction, and by multiplication by any number. We now examine the steps in Euclid’s algorithm, in the light of this concept. The numbers a and b themselves are certainly linearly dependent on a and b, since a = a(b + 1) ? b(a), b = a(b) ? b(a ? 1). The ? rst equation of the algorithm was a = qb + c.Since b is linearly dependent on a and b, so is qb, and since a is also linearly dependent on a and b, so is a ? qb, that is c. Now the next equation of the algorithm allows us to deduce in the same way that d is linearly dependent on a and b, and so on until we come to the last remainder, which is h. This proves that h is linearly dependent on a and b, as asserted. As an illustration, take the same example as was used in  §6, namely a = 7200 and b = 3132. We work through the equations one at a time, using them to express each remainder in terms of a and b. The ? rst equation was 7200 = 2 ? 3132 + 936, which tells s that 936 = a ? 2b. The second equation was 3 132 = 3 ? 936 + 324, which gives 324 = b ? 3(a ? 2b) = 7b ? 3a. The third equation was 936 = 2 ? 324 + 288, which gives 288 = (a ? 2b) ? 2(7b ? 3a) = 7a ? 16b. The fourth equation was 324 = 1 ? 288 + 36, Factorization and the Primes which gives 36 = (7b ? 3a) ? (7a ? 16b) = 23b ? 10a. 21 This expresses the highest common factor, 36, as the difference of two multiples of the numbers a and b. If one prefers an expression in which the multiple of a comes ? rst, this can be obtained by arguing that 23b ? 10a = (M ? 10)a ? (N ? 23)b, provided that Ma = N b.Since a and b have the common factor 36, this factor can be removed from both of them, and the condition on M and N becomes 200M = 87N . The simplest choice for M and N is M = 87, N = 200, which on substitution gives 36 = 77a ? 177b. Returning to the general theory, we can express the result in another form. Suppose a, b, n are given natural numbers, and it is desired to ? nd natural numbers x and y such that ax ? by = n. (6) Such an e quation is called an indeterminate equation since it does not determine x and y completely, or a Diophantine equation after Diophantus of Alexandria (third century A . D . , who wrote a famous treatise on arithmetic. The equation (6) cannot be soluble unless n is a multiple of the highest common factor h of a and b; for this highest common factor divides ax ? by, whatever values x and y may have. Now suppose that n is a multiple of h, say, n = mh. Then we can solve the equation; for all we have to do is ? rst solve the equation ax1 ? by1 = h, as we have seen how to do above, and then multiply throughout by m, getting the solution x = mx1 , y = my1 for the equation (6). Hence the linear indeterminate equation (6) is soluble in natural numbers x, y if and only if n is a multiple of h.In particular, if a and b are relatively prime, so that h = 1, the equation is soluble whatever value n may have. As regards the linear indeterminate equation ax + by = n, we have found the condition for it to be soluble, not in natural numbers, but in integers of opposite signs: one positive and one negative. The question of when this equation is soluble in natural numbers is a more dif? cult one, and one that cannot well be completely answered in any simple way. Certainly 22 The Higher Arithmetic n must be a multiple of h, but also n must not be too small in relation to a and b.It can be proved quite easily that the equation is soluble in natural numbers if n is a multiple of h and n > ab. 9. Factorizing a number The obvious way of factorizing a number is to test whether it is divisible by 2 or by 3 or by 5, and so on, using the series of primes. If a number N v is not divisible by any prime up to N , it must be itself a prime; for any composite number has at least two prime factors, and they cannot both be v greater than N . The process is a very laborious one if the number is at all large, and for this reason factor tables have been computed.The most extensive one which is gener ally accessible is that of D. N. Lehmer (Carnegie Institute, Washington, Pub. No. 105. 1909; reprinted by Hafner Press, New York, 1956), which gives the least prime factor of each number up to 10,000,000. When the least prime factor of a particular number is known, this can be divided out, and repetition of the process gives eventually the complete factorization of the number into primes. Several mathematicians, among them Fermat and Gauss, have invented methods for reducing the amount of trial that is necessary to factorize a large number.Most of these involve more knowledge of number-theory than we can postulate at this stage; but there is one method of Fermat which is in principle extremely simple and can be explained in a few words. Let N be the given number, and let m be the least number for which m 2 > N . Form the numbers m 2 ? N , (m + 1)2 ? N , (m + 2)2 ? N , . . . . (7) When one of these is reached which is a perfect square, we get x 2 ? N = y 2 , and consequently N = x 2 ? y 2 = (x ? y)(x + y). The calculation of the numbers (7) is facilitated by noting that their successive differences increase at a constant rate. The identi? ation of one of them as a perfect square is most easily made by using Barlow’s Table of Squares. The method is particularly successful if the number N has a factorization in which the two factors are of about the same magnitude, since then y is small. If N is itself a prime, the process goes on until we reach the solution provided by x + y = N , x ? y = 1. As an illustration, take N = 9271. This comes between 962 and 972 , so that m = 97. The ? rst number in the series (7) is 972 ? 9271 = 138. The Factorization and the Primes 23 subsequent ones are obtained by adding successively 2m + 1, then 2m + 3, and so on, that is, 195, 197, and so on.This gives the series 138, 333, 530, 729, 930, . . . . The fourth of these is a perfect square, namely 272 , and we get 9271 = 1002 ? 272 = 127 ? 73. An interesting algorithm for fact orization has been discovered recently by Captain N. A. Draim, U . S . N. In this, the result of each trial division is used to modify the number in preparation for the next division. There are several forms of the algorithm, but perhaps the simplest is that in which the successive divisors are the odd numbers 3, 5, 7, 9, . . . , whether prime or not. To explain the rules, we work a numerical example, say N = 4511. The ? st step is to divide by 3, the quotient being 1503 and the remainder 2: 4511 = 3 ? 1503 + 2. The next step is to subtract twice the quotient from the given number, and then add the remainder: 4511 ? 2 ? 1503 = 1505, 1505 + 2 = 1507. The last number is the one which is to be divided by the next odd number, 5: 1507 = 5 ? 301 + 2. The next step is to subtract twice the quotient from the ? rst derived number on the previous line (1505 in this case), and then add the remainder from the last line: 1505 ? 2 ? 301 = 903, 903 + 2 = 905. This is the number which is to be divi ded by the next odd number, 7. Now we an continue in exactly the same way, and no further explanation will be needed: 905 = 7 ? 129 + 2, 903 ? 2 ? 129 = 645, 645 ? 2 ? 71 = 503, 503 ? 2 ? 46 = 411, 645 + 2 = 647, 503 + 8 = 511, 411 + 5 = 416, 647 = 9 ? 71 + 8, 511 = 11 ? 46 + 5, 416 = 13 ? 32 + 0. 24 The Higher Arithmetic We have reached a zero remainder, and the algorithm tells us that 13 is a factor of the given number 4511. The complementary factor is found by carrying out the ? rst half of the next step: 411 ? 2 ? 32 = 347. In fact 4511 = 13? 347, and as 347 is a prime the factorization is complete. To justify the algorithm generally is a matter of elementary algebra.Let N1 be the given number; the ? rst step was to express N1 as N1 = 3q1 + r1 . The next step was to form the numbers M2 = N1 ? 2q1 , The number N2 was divided by 5: N2 = 5q2 + r2 , and the next step was to form the numbers M3 = M2 ? 2q2 , N 3 = M3 + r 2 , N 2 = M2 + r 1 . and so the process was continued. It can be deduced from these equations that N2 = 2N1 ? 5q1 , N3 = 3N1 ? 7q1 ? 7q2 , N4 = 4N1 ? 9q1 ? 9q2 ? 9q3 , and so on. Hence N2 is divisible by 5 if and only if 2N1 is divisible by 5, or N1 divisible by 5. Again, N3 is divisible by 7 if and only if 3N1 is divisible by 7, or N1 divisible by 7, and so on.When we reach as divisor the least prime factor of N1 , exact divisibility occurs and there is a zero remainder. The general equation analogous to those given above is Nn = n N1 ? (2n + 1)(q1 + q2 +  ·  ·  · + qn? 1 ). The general equation for Mn is found to be Mn = N1 ? 2(q1 + q2 +  ·  ·  · + qn? 1 ). (9) If 2n + 1 is a factor of the given number N1 , then Nn is exactly divisible by 2n + 1, and Nn = (2n + 1)qn , whence n N1 = (2n + 1)(q1 + q2 +  ·  ·  · + qn ), (8) Factorization and the Primes by (8). Under these circumstances, we have, by (9), Mn+1 = N1 ? 2(q1 + q2 +  ·  ·  · + qn ) = N1 ? 2 n 2n + 1 N1 = N1 . n + 1 25 Thus the complementary factor to the factor 2n + 1 is Mn+1 , as stated in the example. In the numerical example worked out above, the numbers N1 , N2 , . . . decrease steadily. This is always the case at the beginning of the algorithm, but may not be so later. However, it appears that the later numbers are always considerably less than the original number. 10. The series of primes Although the notion of a prime is a very natural and obvious one, questions concerning the primes are often very dif? cult, and many such questions are quite unanswerable in the present state of mathematical knowledge.We conclude this chapter by mentioning brie? y some results and conjectures about the primes. In  §3 we gave Euclid’s proof that there are in? nitely many primes. The same argument will also serve to prove that there are in? nitely many primes of certain speci? ed forms. Since every prime after 2 is odd, each of them falls into one of the two progressions (a) 1, 5, 9, 13, 17, 21, 25, . . . , (b) 3, 7, 11, 15, 19, 23, 27, . . . ; the progression (a) consisting of all numbers of the form 4x + 1, and the progression (b) of all numbers of the form 4x ? 1 (or 4x + 3, which comes to the same thing).We ? rst prove that there are in? nitely many primes in the progression (b). Let the primes in (b) be enumerated as q1 , q2 , . . . , beginning with q1 = 3. Consider the number N de? ned by N = 4(q1 q2 . . . qn ) ? 1. This is itself a number of the form 4x ? 1. Not every prime factor of N can be of the form 4x + 1, because any product of numbers which are all of the form 4x + 1 is itself of that form, e. g. (4x + 1)(4y + 1) = 4(4x y + x + y) + 1. Hence the number N has some prime factor of the form 4x ? 1. This cannot be any of the primes q1 , q2 , . . . , qn , since N leaves the remainder ? when 26 The Higher Arithmetic divided by any of them. Thus there exists a prime in the series (b) which is different from any of q1 , q2 , . . . , qn ; and this proves the proposition. The same argument cannot be used to prove t hat there are in? nitely many primes in the series (a), because if we construct a number of the form 4x +1 it does not follow that this number will necessarily have a prime factor of that form. However, another argument can be used. Let the primes in the series (a) be enumerated as r1 , r2 , . . . , and consider the number M de? ned by M = (r1 r2 . . rn )2 + 1. We shall see later (III. 3) that any number of the form a 2 + 1 has a prime factor of the form 4x + 1, and is indeed entirely composed of such primes, together possibly with the prime 2. Since M is obviously not divisible by any of the primes r1 , r2 , . . . , rn , it follows as before that there are in? nitely many primes in the progression (a). A similar situation arises with the two progressions 6x + 1 and 6x ? 1. These progressions exhaust all numbers that are not divisible by 2 or 3, and therefore every prime after 3 falls in one of these two progressions.One can prove by methods similar to those used above that there ar e in? nitely many primes in each of them. But such methods cannot cope with the general arithmetical progression. Such a progression consists of all numbers ax +b, where a and b are ? xed and x = 0, 1, 2, . . . , that is, the numbers b, b + a, b + 2a, . . . . If a and b have a common factor, every number of the progression has this factor, and so is not a prime (apart from possibly the ? rst number b). We must therefore suppose that a and b are relatively prime. It then seems plausible that the progression will contain in? itely many primes, i. e. that if a and b are relatively prime, there are in? nitely many primes of the form ax + b. Legendre seems to have been the ? rst to realize the importance of this proposition. At one time he thought he had a proof, but this turned out to be fallacious. The ? rst proof was given by Dirichlet in an important memoir which appeared in 1837. This proof used analytical methods (functions of a continuous variable, limits, and in? nite series), an d was the ? rst really important application of such methods to the theory of numbers.It opened up completely new lines of development; the ideas underlying Dirichlet’s argument are of a very general character and have been fundamental for much subsequent work applying analytical methods to the theory of numbers. Factorization and the Primes 27 Not much is known about other forms which represent in? nitely many primes. It is conjectured, for instance, that there are in? nitely many primes of the form x 2 + 1, the ? rst few being 2, 5, 17, 37, 101, 197, 257, . . . . But not the slightest progress has been made towards proving this, and the question seems hopelessly dif? cult.Dirichlet did succeed, however, in proving that any quadratic form in two variables, that is, any form ax 2 + bx y + cy 2 , in which a, b, c are relatively prime, represents in? nitely many primes. A question which has been deeply investigated in modern times is that of the frequency of occurrence of the p rimes, in other words the question of how many primes there are among the numbers 1, 2, . . . , X when X is large. This number, which depends of course on X , is usually denoted by ? (X ). The ? rst conjecture about the magnitude of ? (X ) as a function of X seems to have been made independently by Legendre and Gauss about X 1800.It was that ? (X ) is approximately log X . Here log X denotes the natural (so-called Napierian) logarithm of X , that is, the logarithm of X to the base e. The conjecture seems to have been based on numerical evidence. For example, when X is 1,000,000 it is found that ? (1,000,000) = 78,498, whereas the value of X/ log X (to the nearest integer) is 72,382, the ratio being 1. 084 . . . . Numerical evidence of this kind may, of course, be quite misleading. But here the result suggested is true, in the sense that the ratio of ? (X ) to X/ log X tends to the limit 1 as X tends to in? ity. This is the famous Prime Number Theorem, ? rst proved by Hadamard and de la Vall? e e Poussin independently in 1896, by the use of new and powerful analytical methods. It is impossible to give an account here of the many other results which have been proved concerning the distribution of the primes. Those proved in the nineteenth century were mostly in the nature of imperfect approaches towards the Prime Number Theorem; those of the twentieth century included various re? nements of that theorem. There is one recent event to which, however, reference should be made.We have already said that the proof of Dirichlet’s Theorem on primes in arithmetical progressions and the proof of the Prime Number Theorem were analytical, and made use of methods which cannot be said to belong properly to the theory of numbers. The propositions themselves relate entirely to the natural numbers, and it seems reasonable that they should be provable without the intervention of such foreign ideas. The search for ‘elementary’ proofs of these two theorems was u nsuccessful until fairly recently. In 1948 A. Selberg found the ? rst elementary proof of Dirichlet’s Theorem, and with 28 The Higher Arithmetic he help of P. Erd? s he found the ? rst elementary proof of the Prime Numo ber Theorem. An ‘elementary’ proof, in this connection, means a proof which operates only with natural numbers. Such a proof is not necessarily simple, and indeed both the proofs in question are distinctly dif? cult. Finally, we may mention the famous problem concerning primes which was propounded by Goldbach in a letter to Euler in 1742. Goldbach suggested (in a slightly different wording) that every even number from 6 onwards is representable as the sum of two primes other than 2, e. g. 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 = 5 + 5, 12 = 5 + 7, . . . Any problem like this which relates to additive properties of primes is necessarily dif? cult, since the de? nition of a prime and the natural properties of primes are all expressed in terms of multiplic ation. An important contribution to the subject was made by Hardy and Littlewood in 1923, but it was not until 1930 that anything was rigorously proved that could be considered as even a remote approach towards a solution of Goldbach’s problem. In that year the Russian mathematician Schnirelmann proved that there is some number N such that every number from some point onwards is representable as the sum of at most N primes.A much nearer approach was made by Vinogradov in 1937. He proved, by analytical methods of extreme subtlety, that every odd number from some point onwards is representable as the sum of three primes. This was the starting point of much new work on the additive theory of primes, in the course of which many problems have been solved which would have been quite beyond the scope of any pre-Vinogradov methods. A recent result in connection with Goldbach’s problem is that every suf? ciently large even number is representable as the sum of two numbers, one of which is a prime and the other of which has at most two prime factors.Notes Where material is changing more rapidly than print cycles permit, we have chosen to place some of the material on the book’s website: www. cambridge. org/davenport. Symbols such as  ¦I:0 are used to indicate where there is such additional material.  §1. The main dif? culty in giving any account of the laws of arithmetic, such as that given here, lies in deciding which of the various concepts should come ? rst. There are several possible arrangements, and it seems to be a matter of taste which one prefers. It is no part of our purpose to analyse further the concepts and laws of ? rithmetic. We take the commonsense (or na? ve) view that we all ‘know’ Factorization and the Primes 29 the natural numbers, and are satis? ed of the validity of the laws of arithmetic and of the principle of induction. The reader who is interested in the foundations of mathematics may consult Bertrand Russe ll, Introduction to Mathematical Philosophy (Allen and Unwin, London), or M. Black, The Nature of Mathematics (Harcourt, Brace, New York). Russell de? nes the natural numbers by selecting them from numbers of a more general kind. These more general numbers are the (? ite or in? nite) cardinal numbers, which are de? ned by means of the more general notions of ‘class’ and ‘one-to-one correspondence’. The selection is made by de? ning the natural numbers as those which possess all the inductive properties. (Russell, loc. cit. , p. 27). But whether it is reasonable to base the theory of the natural numbers on such a vague and unsatisfactory concept as that of a class is a matter of opinion. ‘Dolus latet in universalibus’ as Dr Johnson remarked.  §2. The objection to using the principle of induction as a de? ition of the natural numbers is that it involves references to ‘any proposition about a natural number n’. It seems plain the th at ‘propositions’ envisaged here must be statements which are signi? cant when made about natural numbers. It is not clear how this signi? cance can be tested or appreciated except by one who already knows the natural numbers.  §4. I am not aware of having seen this proof of the uniqueness of prime factorization elsewhere, but it is unlikely that it is new. For other direct proofs, see Mathews, p. 2, or Hardy and Wright, p. 21.?  §5. It has been shown by (intelligent! computer searches that there is no odd perfect number less than 10300 . If an odd perfect number exists, it has at least eight distinct prime factors, of which the largest exceeds 108 . For references and other information on perfect or ‘nearly perfect’ numbers, see Guy, sections A. 3, B. 1 and B. 2.  ¦I:1  §6. A critical reader may notice that in two places in this section I have used principles that were not explicitly stated in  §Ã‚ §1 and 2. In each place, a proof by induction co uld have been given, but to have done so would have distracted the reader’s attention from the main issues.The question of the length of Euclid’s algorithm is discussed in Uspensky and Heaslet, ch. 3, and D. E. Knuth’s The Art of Computer Programming vol. II: Seminumerical Algorithms (Addison Wesley, Reading, Mass. , 3rd. ed. , 1998) section 4. 5. 3.  §9. For an account of early methods of factoring, see Dickson’s History Vol. I, ch. 14. For a discussion of the subject as it appeared in ? Particulars of books referred to by their authors’ names will be found in the Bibliography. 30 The Higher Arithmetic the 1970s see the article by Richard K. Guy, ‘How to factor a number’, Congressus Numerantium XVI Proc. th Manitoba Conf. Numer. Math. , Winnipeg, 1975, 49–89, and at the turn of the millennium see Richard P. Brent, ‘Recent progress and prospects for integer factorisation algorithms’, Springer Lecture Notes in Comp uter Science 1858 Proc. Computing and Combinatorics, 2000, 3–22. The subject is discussed further in Chapter VIII. It is doubtful whether D. N. Lehmer’s tables will ever be extended, since with them and a pocket calculator one can easily check whether a 12-digit number is a prime. Primality testing is discussed in VIII. 2 and VIII. 9. For Draim’s algorithm, see Mathematics Magazine, 25 (1952) 191–4. 10. An excellent account of the distribution of primes is given by A. E. Ingham, The Distribution of Prime Numbers (Cambridge Tracts, no. 30, 1932; reprinted by Hafner Press, New York, 1971). For a more recent and extensive account see H. Davenport, Multiplicative Number Theory, 3rd. ed. (Springer, 2000). H. Iwaniec (Inventiones Math. 47 (1978) 171–88) has shown that for in? nitely many n the number n 2 + 1 is either prime or the product of at most two primes, and indeed the same is true for any irreducible an 2 + bn + c with c odd. Dirichlet’s p roof of his theorem (with a modi? ation due to Mertens) is given as an appendix to Dickson’s Modern Elementary Theory of Numbers. An elementary proof of the Prime Number Theorem is given in ch. 22 of Hardy and Wright. An elementary proof of the asymptotic formula for the number of primes in an arithmetic progression is given in Gelfond and Linnik, ch. 3. For a survey of early work on Goldbach’s problem, see James, Bull. American Math. Soc. , 55 (1949) 246–60. It has been veri? ed that every even number from 6 to 4 ? 1014 is the sum of two primes, see Richstein, Math. Comp. , 70 (2001) 1745–9. For a proof of Chen’s theorem that every suf? iently large even integer can be represented as p + P2 , where p is a prime, and P2 is either a prime or the product of two primes, see ch. 11 of Sieve Methods by H. Halberstam and H. E. Richert (Academic Press, London, 1974). For a proof of Vinogradov’s result, see T. Estermann, Introduction to Modern Prime Number Theory (Cambridge Tracts, no. 41, 1952) or H. Davenport, Multiplicative Number Theory, 3rd. ed. (Springer, 2000). ‘Suf? ciently large’ in Vinogradov’s result has now been quanti? ed as ‘greater than 2 ? 101346 ’, see M. -C. Liu and T. Wang, Acta Arith. , 105 (2002) 133–175.Conversely, we know that it is true up to 1. 13256 ? 1022 (Ramar? and Saouter in J. Number Theory 98 (2003) 10–33). e II CONGRUENCES 1. The congruence notation It often happens that for the purposes of a particular calculation, two numbers which differ by a multiple of some ? xed number are equivalent, in the sense that they produce the same result. For example, the value of (? 1)n depends only on whether n is odd or even, so that two values of n which differ by a multiple of 2 give the same result. Or again, if we are concerned only with the last digit of a number, then for that purpose two umbers which differ by a multiple of 10 are effectively the same. The congruence notation, introduced by Gauss, serves to express in a convenient form the fact that two integers a and b differ by a multiple of a ? xed natural number m. We say that a is congruent to b with respect to the modulus m, or, in symbols, a ? b (mod m). The meaning of this, then, is simply that a ? b is divisible by m. The notation facilitates calculations in which numbers differing by a multiple of m are effectively the same, by stressing the analogy between congruence and equality.Congruence, in fact, means ‘equality except for the addition of some multiple of m’. A few examples of valid congruences are: 63 ? 0 (mod 3), 7 ? ?1 (mod 8), 52 ? ?1 (mod 13). A congruence to the modulus 1 is always valid, whatever the two numbers may be, since every number is a multiple of 1. Two numbers are congruent with respect to the modulus 2 if they are of the same parity, that is, both even or both odd. 31 32 The Higher Arithmetic Two congruences can be added, subtracted, or m ultiplied together, in just the same way as two equations, provided all the congruences have the same modulus.If a ? ? (mod m) and b ? ? (mod m) then a + b ? ? + ? (mod m), a ? b ? ? ? ? (mod m), ab ? (mod m). The ? rst two of these statements are immediate; for example (a + b) ? (? + ? ) is a multiple of m because a ? ? and b ? ? are both multiples of m. The third is not quite so immediate and is best proved in two steps. First ab ? ?b because ab ? ?b = (a ? ?)b, and a ? ? is a multiple of m. Next, ? b ? , for a similar reason. Hence ab ? (mod m). A congruence can always be multiplied throughout by any integer: if a ? ? (mod m) then ka ? k? (mod m).Indeed this is a special case of the third result above, where b and ? are both k. But it is not always legitimate to cancel a factor from a congruence. For example 42 ? 12 (mod 10), but it is not permissible to cancel the factor 6 from the numbers 42 and 12, since this would give the false result 7 ? 2 (mod 10). The reason is obvious : the ? rst congruence states that 42 ? 12 is a multiple of 10, but this does not imply that 1 (42 ? 12) is a multiple of 10. The cancellation of 6 a factor from a congruence is legitimate if the factor is relatively prime to the modulus.For let the given congruence be ax ? ay (mod m), where a is the factor to be cancelled, and we suppose that a is relatively prime to m. The congruence states that a(x ? y) is divisible by m, and it follows from the last proposition in I. 5 that x ? y is divisible by m. An illustration of the use of congruences is provided by the well-known rules for the divisibility of a number by 3 or 9 or 11. The usual representation of a number n by digits in the scale of 10 is really a representation of n in the form n = a + 10b + 100c +  ·  ·  · , where a, b, c, . . . re the digits of the number, read from right to left, so that a is the number of units, b the number of tens, and so on. Since 10 ? 1 (mod 9), we have also 102 ? 1 (mod 9), 103 ? 1 (mod 9), and so on. Hence it follows from the above representation of n that n ? a + b + c +  ·  ·  · (mod 9). Congruences 33 In other words, any number n differs from the sum of its digits by a multiple of 9, and in particular n is divisible by 9 if and only if the sum of its digits is divisible by 9. The same applies with 3 in place of 9 throughout. The rule for 11 is based on the fact that 10 ? ?1 (mod 11), so that 102 ? +1 (mod 11), 103 ? 1 (mod 11), and so on. Hence n ? a ? b + c ?  ·  ·  · (mod 11). It follows that n is divisible by 11 if and only if a ? b+c?  ·  ·  · is divisible by 11. For example, to test the divisibility of 9581 by 11 we form 1? 8+5? 9, or ? 11. Since this is divisible by 11, so is 9581. 2. Linear congruences It is obvious that every integer is congruent (mod m) to exactly one of the numbers 0, 1, 2, . . . , m ? 1. (1) r < m, For we can express the integer in the form qm + r , where 0 and then it is congruent to r (mod m). Obviously there are othe r sets of numbers, besides the set (1), which have the same property, e. . any integer is congruent (mod 5) to exactly one of the numbers 0, 1, ? 1, 2, ? 2. Any such set of numbers is said to constitute a complete set of residues to the modulus m. Another way of expressing the de? nition is to say that a complete set of residues (mod m) is any set of m numbers, no two of which are congruent to one another. A linear congruence, by analogy with a linear equation in elementary algebra, means a congruence of the form ax ? b (mod m). (2) It is an important fact that any such congruence is soluble for x, provided that a is relatively prime to m.The simplest way of proving this is to observe that if x runs through the numbers of a complete set of residues, then the corresponding values of ax also constitute a complete set of residues. For there are m of these numbers, and no two of them are congruent, since ax 1 ? ax2 (mod m) would involve x1 ? x2 (mod m), by the cancellation of the factor a (permissible since a is relatively prime to m). Since the numbers ax form a complete set of residues, there will be exactly one of them congruent to the given number b. As an example, consider the congruence 3x ? 5 (mod 11). 34 The Higher ArithmeticIf we give x the values 0, 1, 2, . . . , 10 (a complete set of residues to the modulus 11), 3x takes the values 0, 3, 6, . . . , 30. These form another complete set of residues (mod 11), and in fact they are congruent respectively to 0, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8. The value 5 occurs when x = 9, and so x = 9 is a solution of the congruence. Naturally any number congruent to 9 (mod 11) will also satisfy the congruence; but nevertheless we say that the congruence has one solution, meaning that there is one solution in any complete set of residues. In other words, all solutions are mutually congruent.The same applies to the general congruence (2); such a congruence (provided a is relatively prime to m) is precisely equivalent to the con gruence x ? x0 (mod m), where x0 is one particular solution. There is another way of looking at the linear congruence (2). It is equivalent to the equation ax = b + my, or ax ? my = b. We proved in I. 8 that such a linear Diophantine equation is soluble for x and y if a and m are relatively prime, and that fact provides another proof of the solubility of the linear congruence. But the proof given above is simpler, and illustrates the advantages gained by using the congruence notation.The fact that the congruence (2) has a unique solution, in the sense explained above, suggests that one may use this solution as an interpretation b for the fraction a to the modulus m. When we do this, we obtain an arithmetic (mod m) in which addition, subtraction and multiplication are always possible, and division is also possible provided that the divisor is relatively prime to m. In this arithmetic there are only a ? nite number of essentially distinct numbers, namely m of them, since two numbers w hich are mutually congruent (mod m) are treated as the same.If we take the modulus m to be 11, as an illustration, a few examples of ‘arithmetic mod 11’ are: 5 ? 9 ? ?2. 3 Any relation connecting integers or fractions in the ordinary sense remains true when interpreted in this arithmetic. For example, the relation 5 + 7 ? 1, 5 ? 6 ? 8, 1 2 7 + = 2 3 6 becomes (mod 11) 6 + 8 ? 3, because the solution of 2x ? 1 is x ? 6, that of 3x ? 2 is x ? 8, and that of 6x ? 7 is x ? 3. Naturally the interpretation given to a fraction depends on the modulus, for instance 2 ? 8 (mod 11), but 2 ? 3 (mod 7). The 3 3 Congruences 35 nly limitation on such calculations is that just mentioned, namely that the denominator of any fraction must be relatively prime to the modulus. If the modulus is a prime (as in the above examples with 11), the limitation takes the very simple form that the denominator must not be congruent to 0 (mod m), and this is exactly analogous to the limitation in ordina ry arithmetic that the denominator must not be equal to 0. We shall return to this point later ( §7). 3. Fermat’s theorem The fact that there are only a ? nite number of essentially different numbers in arithmetic to a modulus m means that there are algebraic relations which are satis? d by every number in that arithmetic. There is nothing analogous to these relations in ordinary arithmetic. Suppose we take any number x and consider its powers x, x 2 , x 3 , . . . . Since there are only a ? nite number of possibilities for these to the modulus m, we must eventually come to one which we have met before, say x h ? x k (mod m), where k < h. If x is relatively prime to m, the factor x k can be cancelled, and it follows that x l ? 1 (mod m), where l ? h ? k. Hence every number x which is relatively prime to m satis? es some congruence of this form. The least exponent l for which x l ? (mod m) will be called the order of x to the modulus m. If x is 1, its order is obviously 1. To illustrate the de? nition, let us calculate the orders of a few numbers to the modulus 11. The powers of 2, taken to the modulus 11, are 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, . . . . Each one is twice the preceding one, with 11 or a multiple of 11 subtracted where necessary to make the result less than 11. The ? rst power of 2 which is ? 1 is 210 , and so the order of 2 (mod 11) is 10. As another example, take the powers of 3: 3, 9, 5, 4, 1, 3, 9, . . . . The ? rst power of 3 which is ? 1 is 35 , so the order of 3 (mod 11) is 5.It will be found that the order of 4 is again 5, and so also is that of 5. It will be seen that the successive powers of x are periodic; when we have reached the ? rst number l for which x l ? 1, then x l+1 ? x and the previous cycle is repeated. It is plain that x n ? 1 (mod m) if and only if n is a multiple of the order of x. In the last example, 3n ? 1 (mod 11) if and only if n is a multiple of 5. This remains valid if n is 0 (since 30 = 1), and it remains valid also for negative exponents, provided 3? n , or 1/3n , is interpreted as a fraction (mod 11) in the way explained in  §2. 36 The Higher ArithmeticIn fact, the negative powers of 3 (mod 11) are obtained by prolonging the series backwards, and the table of powers of 3 to the modulus 11 is n =†¦ ?3 ? 2 ? 1 0 1 2 3 4 5 6 . . . 9 5 4 1 3 9 5 4 1 3 †¦ . 3n ? . . . Fermat discovered that if the modulus is a prime, say p, then every integer x not congruent to 0 satis? es x p? 1 ? 1 (mod p). (3) In view of what we have seen above, this is equivalent to saying that the order of any number is a divisor of p ? 1. The result (3) was mentioned by Fermat in a letter to Fr? nicle de Bessy of 18 October 1640, in which he e also stated that he had a proof.But as with most of Fermat’s discoveries, the proof was not published or preserved. The ? rst known proof seems to have been given by Leibniz (1646–1716). He proved that x p ? x (mod p), which is equivalent to (3), b y writing x as a sum 1 + 1 +  ·  ·  · + 1 of x units (assuming x positive), and then expanding (1 + 1 +  ·  ·  · + 1) p by the multinomial theorem. The terms 1 p + 1 p +  ·  ·  · + 1 p give x, and the coef? cients of all the other terms are easily proved to be divisible by p. Quite a different proof was given by Ivory in 1806. If x ? 0 (mod p), the integers x, 2x, 3x, . . . , ( p ? )x are congruent (in some order) to the numbers 1, 2, 3, . . . , p ? 1. In fact, each of these sets constitutes a complete set of residues except that 0 has been omitted from each. Since the two sets are congruent, their products are congruent, and so (x)(2x)(3x) . . . (( p ? 1)x) ? (1)(2)(3) . . . ( p ? 1)(mod p). Cancelling the factors 2, 3, . . . , p ? 1, as is permissible, we obtain (3). One merit of this proof is that it can be extended so as to apply to the more general case when the modulus is no longer a prime. The generalization of the result (3) to any modulus was ? rst given b y Euler in 1760.To formulate it, we must begin by considering how many numbers in the set 0, 1, 2, . . . , m ? 1 are relatively prime to m. Denote this number by ? (m). When m is a prime, all the numbers in the set except 0 are relatively prime to m, so that ? ( p) = p ? 1 for any prime p. Euler’s generalization of Fermat’s theorem is that for any modulus m, x ? (m) ? 1 (mod m), provided only that x is relatively prime to m. (4) Congruences 37 To prove this, it is only necessary to modify Ivory’s method by omitting from the numbers 0, 1, . . . , m ? 1 not only the number 0, but all numbers which are not relatively prime to m.There remain ? (m) numbers, say a 1 , a2 , . . . , a? , Then the numbers a1 x, a2 x, . . . , a? x are congruent, in some order, to the previous numbers, and on multiplying and cancelling a1 , a2 , . . . , a? (as is permissible) we obtain x ? ? 1 (mod m), which is (4). To illustrate this proof, take m = 20. The numbers less than 20 and relati vely prime to 20 are 1, 3, 7, 9, 11, 13, 17, 19, so that ? (20) = 8. If we multiply these by any number x which is relatively prime to 20, the new numbers are congruent to the original numbers in some other order.For example, if x is 3, the new numbers are congruent respectively to 3, 9, 1, 7, 13, 19, 11, 17 (mod 20); and the argument proves that 38 ? 1 (mod 20). In fact, 38 = 6561. where ? = ? (m). 4. Euler’s function ? (m) As we have just seen, this is the number of numbers up to m that are relatively prime to m. It is natural to ask what relation ? (m) bears to m. We saw that ? ( p) = p ? 1 for any prime p. It is also easy to evaluate ? ( p a ) for any prime power pa . The only numbers in the set 0, 1, 2, . . . , pa ? 1 which are not relatively prime to p are those that are divisible by p. These are the numbers pt, where t = 0, 1, . . , pa? 1 ? 1. The number of them is pa? 1 , and when we subtract this from the total number pa , we obtain ? ( pa ) = pa ? pa? 1 = pa? 1 ( p ? 1). (5) The determination of ? (m) for general values of m is effected by proving that this function is multiplicative. By this is meant that if a and b are any two relatively prime numbers, then ? (ab) = ? (a)? (b). (6) 38 The Higher Arithmetic To prove this, we begin by observing a general principle: if a and b are relatively prime, then two simultaneous congruences of the form x ? ? (mod a), x ? ? (mod b) (7) are precisely equivalent to one congruence to the modulus ab.For the ? rst congruence means that x = ? + at where t is an integer. This satis? es the second congruence if and only if ? + at ? ? (mod b), or at ? ? ? ? (mod b). This, being a linear congruence for t, is soluble. Hence the two congruences (7) are simultaneously soluble. If x and x are two solutions, we have x ? x (mod a) and x ? x (mod b), and therefore x ? x (mod ab). Thus there is exactly one solution to the modulus ab. This principle, which extends at once to several congruences, provided that the moduli ar e relatively prime in pairs, is sometimes called ‘the Chinese remainder theorem’.It assures us of the existence of numbers which leave prescribed remainders on division by the moduli in question. Let us represent the solution of the two congruences (7) by x ? [? , ? ] (mod ab), so that [? , ? ] is a certain number depending on ? and ? (and also on a and b of course) which is uniquely determined to the modulus ab. Different pairs of values of ? and ? give rise to different values for [? , ? ]. If we give ? the values 0, 1, . . . , a ? 1 (forming a complete set of residues to the modulus a) and similarly give ? the values 0, 1, . . . , b ? 1, the resulting values of [? , ? constitute a complete set of residues to the modulus ab. It is obvious that if ? has a factor in common with a, then x in (7) will also have that factor in common with a, in other words, [? , ? ] will have that factor in common with a. Thus [? , ? ] will only be relatively prime to ab if ? is relatively prime to a and ? is relatively prime to b, and conversely these conditions will ensure that [? , ? ] is relatively prime to ab. It follows that if we give ? the ? (a) possible values that are less than a and prime to a, and give ? the ? (b) values that are less than b and prime to b, there result ? (a)? (b) values of [? ? ], and these comprise all the numbers that are less than ab and relatively prime to ab. Hence ? (ab) = ? (a)? (b), as asserted in (6). To illustrate the situation arising in the above proof, we tabulate below the values of [? , ? ] when a = 5 and b = 8. The possible values for ? are 0, 1, 2, 3, 4, and the possible values for ? are 0, 1, 2, 3, 4, 5, 6, 7. Of these there are four values of ? which are relatively prime to a, corresponding to the fact that ? (5) = 4, and four values of ? that are relatively prime to b, Congruences 39 corresponding to the fact that ? (8) = 4, in accordance with the formula (5).These values are italicized, as also are the corresponding values of [? , ? ]. The latter constitute the sixteen numbers that are relatively prime to 40 and less than 40, thus verifying that ? (40) = ? (5)? (8) = 4 ? 4 = 16. ? ? 0 1 2 3 4 0 0 16 32 8 24 1 25 1 17 33 9 2 10 26 2 18 34 3 35 11 27 3 19 4 20 36 12 28 4 5 5 21 37 13 29 6 30 6 22 38 14 7 15 31 7 23 39 We now return to the original question, that of evaluating ? (m) for any number m. Suppose the factorization of m into prime powers is m = pa q b . . . . Then it follows from (5) and (6) that ? (m) = ( pa ? pa? 1 )(q b ? q b? 1 ) . . . or, more elegantly, ? (m) = m 1 ? For example, ? (40) = 40 1 ? and ? (60) = 60 1 ? 1 2 1 2 1 p 1? 1 q †¦. (8) 1? 1 3 1 5 = 16, 1 5 1? 1? = 16. The function ? (m) has a remarkable property, ? rst given by Gauss in his Disquisitiones. It is that the sum of the numbers ? (d), extended over all the divisors d of a number m, is equal to m itself. For example, if m = 12, the divisors are 1, 2, 3, 4, 6, 12, and we have ? (1) + ? (2) + ? (3) + ? (4) + ? (6) + ? (12) = 1 + 1 + 2 + 2 + 2 + 4 = 12. A general proof can be based either on (8), or directly on the de? nition of the function. 40 The Higher ArithmeticWe have already referred (I. 5) to a table of the values of ? (m) for m 10, 000. The same volume contains a table giving those numbers m for which ? (m) assumes a given value up to 2,500. This table shows that, up to that point at least, every value assumed by ? (m) is assumed at least twice. It seems reasonable to conjecture that this is true generally, in other words that for any number m there is another number m such that ? (m ) = ? (m). This has never been proved, and any attempt at a general proof seems to meet with formidable dif? culties. For some special types of numbers the result is easy, e. g. f m is odd, then ? (m) = ? (2m); or again if m is not divisible by 2 or 3 we have ? (3m) = ? (4m) = ? (6m). 5. Wilson’s theorem This theorem was ? rst publis