A Course on Number Theory

A Course on Number Theory Peter J. Cameron ii Preface 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. There is nothing original to me in the notes. The Course was designed by Su- san McKay, and developed by Stephen Donkin, Ian Chiswell, Charles Leedham- Green, and Thomas M uller; I have benefited greatly from Ian Chiswell's notes, which I have followed closely. I am grateful to Mark Walters who stood in for me in the first six lectures of the Course , and whose comments have been very helpful; also to the class tutors, markers, and most of all the students who took the Course , for their comments and support. The original Course was largely based on continued fractions: this technique is very amenable to hand calculation, and can be used to solve Pell's equation, to write an integer as a sum of squares where this is possible, and to classify the indefinite binary quadratic forms.

Theorem 1.2 Any natural number greater than 1 can be written as a product of prime numbers, and this expression is unique apart from re-ordering the factors. Proof We show the existence of a factorisation into primes by induction. Given a natural number n, if n is prime, then it is the product of just one prime. (This

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