For a positive integer m and integers a and b, we say that a is congruent to b modulo m if m. Fermats little theorem if p is a prime number and a is a natural number that is not a multiple of p, then ap1. Next, we present fermats in this section we present three applications of congruences. Number theorists study prime numbers as well as the. This set of notes on number theory was originally written in 1995 for students at the imo. Binomial coefficients, congruences, lecture 3 notes.
This list may not reflect recent changes learn more. This idea will be critical to most of what we do the rest of the term. In these number theory notes pdf, you will study the micro aptitude of understanding aesthetic aspect of mathematical instructions and gear young minds to ponder upon such problems. Some of his famous problems were on number theory, and have also been in. In other words, p is prime if its only factors in the natural numbers are itself and 1, and these factors are different. German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics.
Fermats little theorem theorem fermats little theorem if p is a prime, then for any integer a not divisible by p, ap 1 1 mod p. Introductory algebraic number theory algebraic number theory is a subject that came into being through the attempts of mathematicians to try to prove fermats last theorem and that now has a wealth of applications to diophantine equations, cryptography, factoring, primality testing, and publickey cryptosystems. The second is to help you understand some of the fundamental ideas of number theory. The simplest congruence to solve is the linear congruence, ax bpmod mq. Theorem congruence mod n is an equivalence relation. Corollary we can factor a power ab as some product ap 1 ap 1 ap 1 ac, where c is some small number in fact, c b mod p 1. The problem was that it could not be used directly in any application. He proved the fundamental theorems of abelian class. That is, after a certain point, the sequence begins to repeat. Note that primes are the products with only one factor and 1 is the empty product. Divisibility and modular arithmetic are foundational to number theory, which is the theory of the integers. Also, another objective is to make the students familiar with simple number.
Number theory modular arithmetic stanford university. When we take ab mod p, all the powers of ap 1 cancel, and we just need to compute. In the proof, we focus on the general ideas rather than focusing on the speci cs, and the report is best read in conjunction with. Pdf contributions of srinivasa ramanujan to number theory. Overview of number theory basics university of texas at. This version is particularly suited for proofs involving congruences. Every integer greater than 1 can be written uniquely in the form pe 1 1 p e 2 2 p e k k, where the p i are distinct primes and the e i are positive integers. Directions for writing proofs of universal statements writing proofs is similar to writing a computer program based on a set of specifications. The solutions fall into congruence classes, and there are only a nite number of congruence classes that solve the congruence. A computational introduction to number theory and algebra. One of the fundamental tools of number theory is the congruence. Number theory, in turn, is foundational for cryptography, the science of keeping ones communications and data secure from eavesdropping third. I will not use the language of abstract algebra, however, to the extent that the book does.
Two important theorems about congruences are fermats little theorem and eulers the orem, the. For example, if 5x 7 pmod 12q, then one solution is x 11 since 5 11 7 48. From the elementary theory of numbers it is known that if the congruences. Proofs of basic theorems are presented in an interesting and comprehensive way that can be read and understood even by nonmajors with the exception in the last three chapters where a background in analysis, measure theory and abstract algebra is required. This set of notes partially follows the books treatment. For example, 32 rem 5 is the remainder when 32 is divided by 5, which is 2. In this article we define a function l which will allow us to generalize separately or simultaneously some theorems from numbers theory obtained by. Universityofillinois,1914 thesis submittedinpartialfulfillmentoftherequirementsforthe degreeof masterofarts. The first theorem is wilsons theorem which states that p. Regrettably mathematical and statistical content in pdf files is unlikely to be accessible. But k n is an integer because it is a difference of integers. The following 107 pages are in this category, out of 107 total.
A natural number p is said to be prime if p 1 and, whenever p ab holds for some natural numbers a and b, we have either a p, b 1, or a 1, b p. Finally we discuss the distribution of primes via the prime number theorem and the riemann hypothesis. Linear congruences, chinese remainder theorem, algorithms. The number theory, as such, is less applied in engineering compared to calculus, geometry, etc. Proceedings of the first conference of the canadian number theory association, 8192, 1990. If a and b are integers and there is some integer c such that a bc, then we say that b divides a or is a factor or divisor of a and write ba. The next two theorems show that congruences and equations share many properties. The primitive root theorem identi es all the positive integers n modulo which primitive roots exist. Though it seems that arithmetic on z n is the same as on z, do not be deceived.
These notes1 are written for a supplementary lecture in the number theory course. Vinogradov theorem is discussed, along with some other results in number theory necessary to establish the theorem. Square roots, tonellis algorithm, number of consecutive pairs of squares mod p pdf. The following theorem gives the answer for the number of isomorphisms between a and b. In particular, if p is a prime number, then a is coprime with p for every a such that 0 of congruence relations are the following. Theorem of the day the ramanujan partition congruences let n be a nonnegative integer and let pn denote the number of partitions of n that is, the number of ways to write n as a sum of positive integers. W right, an intro duction to the theory of numbers, oxford. Hence m n equals 2 times an integer, and so by definition of even, m n is even. A congruence of the form \ax\equiv bmod\ m\ where \x\ is an unknown integer is called a linear congruence in one variable.
As a rule, these theories are developed sufficiently far so that the reader can see for. In this case, we expect the solution to be a congruence as well. Every natural number is built, in a unique way, out of prime numbers. If we try all the values from x 1 through x 10, we nd that 53 4 mod 11. This notation, and much of the elementary theory of congruence, is due to the famous german mathematician, carl friedrich gausscertainly the outstanding mathematician of his time, and perhaps the greatest mathematician of all time. The remainder r in the division theorem is denoted n rem d.
Congruence, modular arithmetic, 3 ways to interpret a. In other words, n rem d is the remainder when n is divided by d. Find all numbers of the form 2nl with positive integer n, not exceeding million, which are products of two primes, and prove that if n. The number of elements in a reduced residue system modulo m is denoted by. The study of number theory goes back at least to the ancient greeks, who investigated the. The primitive root theorem philadelphia university. Cyclotomic polynomials, primes congruent to 1 mod n pdf 14. The number theory is a branch of mathematics which is primarily dedicated to the study of integers. The integer mis called the modulus of the congruence. By the fundamental theorem of arithmetic, n is divisible by. But, the number theory, combined with the computational power of modern. Remainder theorem the simplest equation to solve in a basic algebra class is the equation ax b, with solution x b a, provided a.
Overview of number theory basics murat kantarcioglu based on prof. Theorem fundamental theorem of arithmetic every positive integer greater than 1 can be written uniquely as a prime or as the product of its prime factors, written in order of nondecreasing size. Number theory or arithmetic or higher arithmetic in older usage is a branch of pure mathematics devoted primarily to the study of the integers and integervalued functions. Cse 1400 applied discrete mathematics number theory and proofs department of computer sciences college of engineering florida tech spring 2012 problems labeled. A prime number is an integer greater than 1 whose only positive divisors are itself and 1. The regular integers are visualized as lying on a number line, where integers to the left are smaller than integers on the right. One goal is to help you develop independent mathematical thinking skills. By remainder, of course, we mean the unique number r defined by the division algorithm. Every natural number can be written as a product of primes uniquely up to order. But each of these mappings is not an isomorphism from a to b.
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