Number Theory(C)

NUMBER THEORY (C)

Review from Part IA Numbers and Sets: Euclid’s Algorithm, prime numbers, fundamental theorem of arithmetic. Congruences. The theorems of Fermat and Euler. [2]
Chinese remainder theorem. Lagrange’s theorem. Primitive roots to an odd prime power modulus. [3]
The mod-p ﬁeld, quadratic residues and non-residues, Legendre’s symbol. Euler’s criterion. Gauss’ lemma, quadratic reciprocity. [2]
Proof of the law of quadratic reciprocity. The Jacobi symbol. [1]
Binary quadratic forms. Discriminants. Standard form. Representation of primes. [5]
Distribution of the primes. Divergence ofPp p−1. The Riemann zeta-function and Dirichlet series. Statement of the prime number theorem and of Dirichlet’s theorem on primes in an arithmetic progression. Legendre’s formula. Bertrand’s postulate. [4]
Continued fractions. Pell’s equation. [3]
Primality testing. Fermat, Euler and strong pseudo-primes. [2]
Factorization. Fermat factorization, factor bases, the continued-fraction method. Pollard’s method. [2]

Appropriate books

A. Baker A Concise Introduction to the Theory of Numbers. Cambridge University Press 1984

Alan Baker A Comprehensive Course in Number Theory. Cambridge University Press 2012

G.H. Hardy and E.M. Wright An Introduction to the Theory of Numbers. Oxford University Press

N. Koblitz A Course in Number Theory and Cryptography. Springer 1994

T. Nagell Introduction to Number Theory. AMS

H. Davenport The Higher Arithmetic. Cambridge University Press

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