# An Introduction to Number Theory by Graham Everest BSc, PhD, Thomas Ward BSc, MSc, PhD (auth.)

By Graham Everest BSc, PhD, Thomas Ward BSc, MSc, PhD (auth.)

An creation to quantity concept offers an advent to the most streams of quantity conception. beginning with the original factorization estate of the integers, the topic of factorization is revisited numerous occasions in the course of the e-book to demonstrate how the tips passed down from Euclid proceed to reverberate during the subject.

In specific, the publication exhibits how the basic Theorem of mathematics, passed down from antiquity, informs a lot of the educating of recent quantity idea. the result's that quantity conception could be understood, now not as a set of tips and remoted effects, yet as a coherent and interconnected idea.

A variety of diversified ways to quantity concept are provided, and the several streams within the ebook are introduced jointly in a bankruptcy that describes the category quantity formulation for quadratic fields and the recognized conjectures of Birch and Swinnerton-Dyer. the ultimate bankruptcy introduces the various major rules in the back of sleek computational quantity idea and its purposes in cryptography.

Written for graduate and complicated undergraduate scholars of arithmetic, this article is going to additionally entice scholars in cognate matters who desire to be brought to a couple of the most subject matters in quantity theory.

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Extra resources for An Introduction to Number Theory

Sample text

1 We wish to prove that any natural number n has a decomposition n = p1 · · · pr into primes uniquely up to rearrangement of the prime factors. For n = 2, the theorem is clearly true. We proceed by induction. Suppose that the Fundamental Theorem of Arithmetic holds for all natural numbers strictly less than some a > 1. We want to deduce the Fundamental Theorem of Arithmetic for a. Let D = {d | d > 1, d a} denote the set of non-identity divisors of a. The set D is nonempty since it contains a, so it has a smallest element, which we denote p.

7 is related to subtle problems in algebraic number theory; see Ribenboim’s book [125] for a discussion and detailed references. That the positive values of a polynomial in several variables could coincide with the primes is essentially a by-product of Matijaseviˇc’s solution to one of Hilbert’s famous problems. Some of the history and references and two explicit polynomials are given in accessible form in the paper [85] of Jones, Sato, Wada and Wiens. 10 follows a survey paper of Dudley [46]. 9) was ﬁrst proved by Tchebychef [151, Tome I, pp.

25. Prove that n = 561 is a composite number that satisﬁes Fermat’s Little Theorem for every possible base by showing that a560 ≡ 1 modulo 561 for every a, 1 < a < n with gcd(a, 561) = 1. 5. 6 Proving the Fundamental Theorem of Arithmetic 35 are inﬁnitely many Carmichael numbers until 1994, when Alford, Granville, and Pomerance not only proved that there are inﬁnitely many but gave some measure of how many there are asymptotically. The existence of inﬁnitely many Carmichael numbers renders the test based on Fermat’s Little Theorem test too unreliable.