Gaps between primes

This project falls within the EPSRC Digital Economy research area. Many critical computer programs rely on properties of prime numbers, but unfortunately these properties are poorly understood from a theoretical viewpoint. This poses a critical security risk: if primes behaved differently to how algorithms assume, then several vital parts of the digital economy would fail. For example, many cryptographic algorithms require one to find an auxiliary prime of a certain size, and they do this by testing consecutive integers for primality. Whilst we know that we can test an individual number quickly, this would take thousands of years if there were no primes close to our starting point, meaning the whole cryptographic algorithm would hang and security would be lost. We believe there should never be such a long string of consecutive integers which are not prime, but we do not know how to prove this. This is just one example; there are many similar cases where critical day-to-day algorithms rely on assumptions about primes, which in turn can be resolved if we can solve some of the most famous problems in theoretical mathematics. Unfortunately such theoretical questions are notoriously difficult and have been thought about by mathematicians for hundreds of years without resolution. Fortunately, in the past few years several new techniques for studying gaps between primes have been developed, which offer new insights into the primes and these old famous problems.

This project is aimed at improving our understanding of gaps between primes, thereby moving in the direction of solving these critical issues for the Digital economy. It will investigate further the new techniques developed in recent years, building on recent breakthroughs to further our understanding of gaps between primes. This naturally requires an interdisciplinary approach to make progress - the intention is to bring together disparate ideas from combinatorics, functional analysis, analytic number theory, numerical analysis and sieve theory to improve the current techniques for studying gaps between primes. This will therefore require significant novelty by introducing new external ideas to build upon and improve the current techniques, making partial progress towards the overall aims which have significant impacts on the digital economy.

There will be no direct companies or collaborators involved in this project, although it is hoped that links will be made with other world experts in this area to strengthen the outcome and impacts of the research performed.