Libraries and resellers, please contact cust-serv ams. See our librarian page for additional eBook ordering options. Its exposition reflects the most recent scholarship in mathematics and its history. Almost sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity.
Geometric and dynamical arguments provide new insights, and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations e. Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem.
Accompanying each chapter, historical notes curate primary sources and secondary scholarship to trace the development of number theory within and outside the Western tradition. Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory. It is also suitable for mathematicians seeking a fresh perspective on an ancient subject.
To view the author's website for sample syllabi, quizzes, student project ideas, and Python programming tutorials, click here. This book is an introduction to number theory like no other. It covers the standard topics of a first course in number theory from integer division with remainder to representation of integers by quadratic forms. Nearly illustrations elucidate proofs, provide data visualization, and give fresh new insights The page layout is exquisite Each chapter begins with a figure on the left side and text on the right side of a two-page spread.
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History of Arithmetic and Number Theory
About this book Number Theory or arithmetic, as some prefer to call it, is the oldest, purest, liveliest, most elementary yet sophisticated field of mathematics. Show all. Read this book on SpringerLink. Two of the most famous problems in number theory involve Fermat.
Many mathematicians, including Mersenne and Euler, have tried to find a formula that will define all the prime numbers. No one has ever succeeded. Fermat had one of the most famous failures. He thought that if he squared 2 and then raised the square of 2 to a higher power, which he labeled n a whole number , then the results would be nothing but primes. This formula appeared to work until Leonhard Euler proved it wrong.
Fermat wrote his famous Last Theorem in the margin of a book some time in the late s. Mathematicians have been trying to prove or disprove this theorem for centuries. Princeton University professor Andrew J. Wiles apparently had proved it correct but later flaws were found in his proof. By late , Wiles thought the flaws had been solved. Wiles announced in June that he had proved Fermat right. Number theory was labeled the Queen of Mathematics by Gauss. For many years, it was thought to be without many practical applications. That situation has changed significantly in the twentieth century with the rise of computers.
Prime and composite numbers play an important role in modern cryptography or coding systems. Huge volumes of confidential information such as credit card numbers and bank account numbers and large amounts of money are transferred electronically around the world every day, all of which must kept secret. One of the important applications of number theory is keeping secrets. However, once a computer finds out that a number is not prime, it then takes a long time to find out what its factors are, especially if the number is a large composite say digits long. It can take years on a supercomputer to find the prime factors of large composite numbers.
This time gap between finding out if a number is prime and factoring the primes in a composite number is useful to cryptographers.
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To create a security system, they invent numerical codes for the letters and characters of a message. Then they use an encoding algorithm a series of steps to solve a problem to turn a message into a long number. If the message is more than a certain length, say characters, then the cryptography program breaks the message into blocks of characters. Once the message is translated into a number, the program multiplies the number of an encoded message by a certain prime which could be a digit number and by a composite number.
The composite number is the product of two prime numbers, which have been randomly selected and which must be in both the encoding algorithm of the sender and the decoding algorithm of the receiver. The prime numbers making up the composite number are usually. Algorithm — Method for solving a numerical problem consisting of a series of mathematical steps.
Composite number — A number that can be divided into two or more prime numbers in addition to 1 and itself. Congruence — The relationship between two numbers if they have the same remainder when they are divided by a number. Cryptography — The study of creating and breaking secret codes.
Factors — Numbers that are multiplied with other numbers to equal a product.
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In addition, factors are the numbers that result when numbers are divided. For example, 10 and 10 are factors of Modulo — A number by which two other numbers can be divided to give the same remainder.
Prime numbers — Numbers that can only be divided evenly by 1 and themselves. Product — The result of multiplying two or more numbers.
Six multiplied by 7 gives the product of Set — A set is a collection of things called members or elements of the set. In mathematics, the members of a set will often be numbers. Whole numbers — The positive integers: 1, 2, 3, 4, When the message is transmitted from the sender to the receiver, some of the numbers are made public, but the primes that make up the composite number are kept secret.
The decoding algorithm of the authorized person who receives the message only knows them. Anyone who is eavesdropping on the transmission will see many of the numbers, but without the prime numbers from the encoding and decoding programs, it is impossible to decode the message in any reasonable time. Computer cryptography systems are only one application of number theory. Other formulas of number theory allow computer programs to find out many years in advance what days of the week will fall on what dates of the month, so that people can find out well in advance what day of the week Christmas or the Fourth of July will occur.
Many computers have preinstalled internal programs that tell users when they last modified a file down to the second, minute, hour, day of the week, and date of the month. These programs work thanks to the formulas of number theorists. Burton, David M.
Elementary Number Theory. Reid, Constance. Wellesley, MA: A. Peters, Rosen, Kenneth H. Elementary Number Theory and Its Applications. Schroeder, Manfred Robert. Berlin , Germany, and New York : Springer, Stopple, Jeffrey. Cambridge: Cambridge University Press, Weisstein, Eric W. Cite this article Pick a style below, and copy the text for your bibliography. September 24, Retrieved September 24, from Encyclopedia. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia. Number theory is the study of natural, or counting numbers, including prime numbers. One formula, invented by Marin Mersenne is 2p - 1, where p is a prime number. Another formula, invented by Leonhard Euler , generates prime numbers regularly for the series of consecutive numbers from 0 to 15 and then stops.
Pierre de Fermat is one of the most famous number theoreticians in history, but mathematics was only his hobby.