By William Stein

This can be a textbook approximately classical trouble-free quantity concept and elliptic curves. the 1st half discusses straightforward issues corresponding to primes, factorization, persisted fractions, and quadratic kinds, within the context of cryptography, computation, and deep open examine difficulties. the second one half is set elliptic curves, their functions to algorithmic difficulties, and their connections with difficulties in quantity concept akin to Fermat’s final Theorem, the Congruent quantity challenge, and the Conjecture of Birch and Swinnerton-Dyer. The meant viewers of this publication is an undergraduate with a few familiarity with easy summary algebra, e.g. jewelry, fields, and finite abelian teams.

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Extra info for Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach

Example text

21] for a book that gives a detailed exposition of this algorithm. 5. ) whether or not an integer is prime. 3). sage: for p in primes(100): ... if is_prime(2^p - 1): ... print p, 2^p - 1 2 3 3 7 5 31 7 127 13 8191 17 131071 19 524287 31 2147483647 61 2305843009213693951 89 618970019642690137449562111 There is a specialized test for primality of Mersenne numbers called the Lucas-Lehmer test. This remarkably simple algorithm determines provably correctly whether or not a number 2p − 1 is prime. We implement it in a few lines of code and use the Lucas-Lehmer test to check for primality of two Mersenne numbers: sage: ...

15 For what values of n is ϕ(n) odd? 16 (a) Prove that ϕ is multiplicative as follows. Suppose m, n are positive integers and gcd(m, n) = 1. Show that the natural map ψ : Z/mnZ → Z/mZ × Z/nZ is an injective homomorphism of rings, hence bijective by counting, then look at unit groups. (b) Prove conversely that if gcd(m, n) > 1, then the natural map ψ : Z/mnZ → Z/mZ × Z/nZ is not an isomorphism. 17 Seven competitive math students try to share a huge hoard of stolen math books equally between themselves.

7 Find four complete sets of residues modulo 7, where the ith set satisfies the ith condition: (1) nonnegative, (2) odd, (3) even, (4) prime. 9 for divisibility of an integer by 5, 9, and 11, and prove each of these rules using arithmetic modulo a suitable n. ) Define a sequence of decimal integers an as follows: a1 = 0, a2 = 1, and an+2 is obtained by writing the digits of an+1 immediately followed by those of an . For example, a3 = 10, a4 = 101, and a5 = 10110. Determine the n such that an is a multiple of 11, as follows: (a) Find the smallest integer n > 1 such that an is divisible by 11.

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