A remarkable inequality among integer numbers is given. Easily deduced from Fermat's Last Theorem, it would be nevertheless very difficult to establish through other means.

In this thesis we consider three different issues of analytic number theory. Firstly, we investigate how residues modulo q may be expressed as products of small primes. In Chapter 1, we work in the regime in which these primes are less than q, and present some partial results towards an open conjecture of Erdös. In Chapter 2, we consider the kinder regime in which these primes are at most q^C , for some constant C that is greater than 1. Here we reach an explicit version of Linnik's Theorem on the least prime in an arithmetic progression, saving that we replace 'prime' with 'product of exactly three primes'. The results of this chapter are joint with Prof. Olivier Ramaré. The next two chapters concern equidistribution modulo 1, specifically the notion that an infinite set of integers is metric poissonian. This strong notion was introduced by Rudnick and Sarnak around twenty years ago, but more recently it has been linked with concepts from additive combinatorics. In Chapter 3 we study the primes in this context, and prove that the primes do not enjoy the metric poissonian property, a theorem which, in passing, improves upon a certain result of Bourgain. In Chapter 4 we continue the investigation further, adapting arguments of Schmidt to demonstrate that certain random sets of integers, which are nearly as dense as the primes, are metric poissonian after all. The major work of this thesis concerns the study of diophantine inequalities. The use of techniques from Fourier analysis to count the number of solutions to such systems, in primes or in other arithmetic sets of interest, is well developed. Our innovation, following suggestions of Wooley and others, is to utilise the additive-combinatorial notion of Gowers norms. In Chapter 5 we adapt methods of Green and Tao to show that, even in an extremely general framework, Gowers norms control the number of solutions weighted by arbitrary bounded functions. We use this result to demonstrate cancellation of the Möbius function over certain irrational patterns.