Number theory has its roots in the study of the properties of the natural numbers n f1,2,3. Olympiad number theory through challenging problems. Figurate numbers and pythagorean triples feature prominently, of course, but there are also proofs of fermats little and wilsons theorems. In section 6 we discuss an algorithm that implements this \fermat test very e ciently. Example 1 the number 102 has the positive divisors 1, 2, 3, 6, 17, 34, 51, 102, and the number 170 has the positive divisors 1, 2, 5, 10, 17, 34, 85, and 170. Number theory naoki sato 0 preface this set of notes on number theory was originally written in 1995 for students at the imo level. Readable discussions motivate new concepts and theorems before their formal definitions and statements are presented. We also say that a is a divisor of b, or that it is a factor of b. The gcd of two or more numbers is the largest positive number that divides all the numbers that are considered.
Similarly, if a number leaves a remainder 3 when divided by 4, it is said to be 3 modulo 4 number. The euclidean algorithm for computing gcd systematically nds smaller and smaller numbers in sumsm. Traverse all the numbers from min a, b to 1 and check whether the current number divides both a and b. For example, the fundamental theorem of arithmetic is a consequence of the tools we develop in order to find all the integral points on a line in the plane. An integer 1 which is not a prime is called composite. If a b modulo n 1, then a is said to be the multiplicative inverse of b in class modulo n b is said to be the multiplicative inverse of a in class modulo n example. More formally, for a 6 0 we say that divides b if there is. It covers the basic background material that an imo student should be familiar with. The systematic study of number theory was initiated around 300b. For any integer n 2z there is a unique integer r in f0. The notes contain a useful introduction to important topics that need to be addressed in a course in number theory. Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. We are dealing with primes p on the order of 300 digits long, 1024 bits.
Sep 22, 20 problem 2009 pumac number theory, problem a1. We need a lemma which belongs to the theory of quadratic residues see chapter 5. These are the notes of the course mth6128, number theory, which i taught at queen mary, university of london, in the spring semester of 2009. Nuggets of number theory will attract fans of visual thinking, number theory, and surprising connections. An example is checking whether universal product codes upc or international standard book number isbn codes are legitimate.
The numbers 31 and 46 are congruent mod 3 because they di. This course starts at the very beginning covering all of the essential tools and concepts in number theory, and then applying them to computational art, cryptography codebreaking, challenging logic puzzles, understanding infinity, and more. Gauss 17771855 was an infant prodigy and arguably the greatest mathematician of all time if such rankings mean anything. The integers and division primes and greatest common divisor applications applications of number theory. Cryptography hash functions iii however, the function is clearly not onetoone. Modular arithmetic, cryptography, and randomness for hundreds of years, number theory was among the least practical of mathematical disciplines.
Given two positive numbers, a and n, a modulo n abbreviated as a mod n is the remainder of the euclidean division of a by n, where a is the dividend and n is the divisor. We read this as a is congruent to b modulo or mod n. Modular arithmetic and elementary algebra 1 euclids algorithm. Thus it is possible to prove that a number can be factored without actually factoring it. In this chapter we study some basic facts and algorithms in number theory, which have important relevance to modern cryptography. A number is said to be 1 modulo 4 number if it leaves a remainder 1 when divided by 4. Integer a is congruent to integer b modulo m 0, if a and b give the same remainder when divided by m. It is used to encode ramification data for abelian. Mar 01, 2014 gauss 17771855 introduced the word modulus in 1801 in his disquisitiones arithmeticae. Section 1 introduces euclids algorithm, which is used to find the hcf of two integers, and the idea of congruences, mathematical statements used to compare remainders when two integers are each divided by another integer.
Consider next the fundamental problem of factoring an integer into primes. Cyclotomic polynomials, primes congruent to 1 mod n pdf 14. This book contains hundreds of visual explanations of results from elementary number theory. These notes serve as course notes for an undergraduate course in number the ory. Cryptography and computer security cs255 very basic number theory fact sheet part i. The simplest congruence to solve is the linear congruence, ax bpmod mq.
Number theory modular arithmetic stanford university. In mathematics, in the field of algebraic number theory, a modulus plural moduli or cycle, or extended ideal is a formal product of places of a global field i. Integer a iscongruent to integer b modulo m 0, if a and b give. Regrettably mathematical and statistical content in pdf files is unlikely to be accessible using a. For the operation that gives a numbers remainder, see modulo operation. When two elements, x 1 6 x 2 hash to the same value, we call it a collision. A lively introduction with proofs, applications, and stories, is a new book that provides a rigorous yet accessible introduction to elementary number theory along with relevant applications. Multiply 7 with all the integers 0, 1, 14 in class modulo 15. If a,b,m are integers, m 0, we say a and b are equivalent mod m, written a.
Most if not all universities worldwide offer introductory courses in number theory for math majors and in many cases as an elective course. Division is notably absent from the above discussion. Multiply 7 with all the integers 0, 1, 14 in class modulo. The exposition is informal, with a wealth of numerical examples that are analyzed for patterns and used to make conjectures. In computing, the modulo operation finds the remainder after division of one number by another called the modulus of the operation. He laid the modern foundations of algebraic number theory by. This chapter lays the foundations for our study of the theory of numbers by weaving together the themes of prime numbers, integer factorization, and the distribution of primes. Find the multiplicative inverse of 7 in class modulo 15 straightforward approach. For example, here are some problems in number theory that remain unsolved. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. A friendly introduction to number theory is an introductory undergraduate text designed to entice nonmath majors into learning some mathematics, while at the same time teaching them how to think mathematically.
The relation congruence modulo m is 1 reflexive 2 stisymmetric 3 transitive only mth t 4 all of these mathematics. The euclidean algorithm and the method of backsubstitution 4 4. These notes serve as course notes for an undergraduate course in number theory. This free course, introduction to number theory, is a branch of mathematics concerned with the properties of integers. The modulo arithmetic lesson aims to teach students addition and multiplication modulo n. Number theory explore the powers of divisibility, modular arithmetic, and infinity. In computing, the modulo operation finds the remainder after division of one number by.
Gauss said that a is congruent to b modulo n iff n j. Then r is called the residue of n modulo m, and by slight abuse of notation we will refer to it as n mod m. Applications cse235 introduction hash functions pseudorandom numbers representation of integers euclids algorithm c. Number theory is a branch of mathematics concerned with properties of. The chinese remainder theorem the simplest equation to solve in a basic algebra class is the equation ax b, with solution x b a, provided a. Gauss 17771855 introduced the word modulus in 1801 in his disquisitiones arithmeticae. Number theory concerns the former case, and discovers criteria upon which one can decide about divisibility of two integers. Integers modulo \n \ however are visualized as lying on a circle e. In contrast to subjects such as arithmetic and geometry, which proved useful in everyday problems in commerce and architecture, as. One can nd the residue of a number n by taking the remainder when dividing by m. For example, the gcd of 6 and 10 is 2 because it is the largest positive number that can divide both 6 and 10.
The inverse of a modulo m exists and is unique iff a is relatively prime to m. Square roots, tonellis algorithm, number of consecutive pairs of squares mod p pdf. Division given two integers, say a and b, the quotient ba may or may not be an integer e. This book is an introduction to number theory and arithmetic geometry, and the goal of the text is to use geometry as the motivation to prove the main theorems in the book. We will encounter all these types of numbers, and many others, in our excursion through the theory of numbers. The regular integers are visualized as lying on a number line, where integers to the left are smaller than integers on the right. Two integers a and b are said to be congruent modulo 5 if they each.
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