Gaps between primes

该项目属于EPSRC数字经济研究领域。许多关键的计算机程序依赖于素数的性质，但不幸的是，从理论角度很难理解这些性质。这构成了一个严重的安全风险：如果素数的表现与算法假设的不同，那么数字经济的几个关键部分将会崩溃。例如，许多密码算法需要找到特定大小的辅助素数，并且它们通过测试连续整数的素性来实现这一点。虽然我们知道我们可以快速测试单个数字，但如果在我们的起点附近没有素数，这将需要数千年的时间，这意味着整个密码算法将挂起，安全性将会丧失。我们认为永远不应该有这么长的连续整数串不是素数，但我们不知道如何证明这一点。这只是一个例子；还有许多类似的情况，关键的日常算法依赖于关于素数的假设，反过来，如果我们能够解决理论数学中一些最著名的问题，这个问题就可以得到解决。不幸的是，这样的理论问题是出了名的难，数学家们几百年来一直在思考，却没有解决。幸运的是，在过去的几年里，已经发展了几种研究素数之间差距的新技术，这些技术为素数和这些老的著名问题提供了新的见解。这个项目旨在提高我们对素数之间差距的理解，从而朝着解决数字经济这些关键问题的方向前进。它将进一步研究近年来开发的新技术，建立在最近的突破基础上，以进一步了解素数之间的差距。这自然需要跨学科的方法来取得进展-其目的是将组合学、泛函分析、解析数理论、数值分析和筛子理论的不同想法结合在一起，以改进目前研究素数之间差距的技术。因此，这将需要极大的新颖性，通过引入新的外部想法来建立和改进现有技术，在实现对数字经济具有重大影响的总体目标方面取得部分进展。这个项目不会有直接的公司或合作者参与，尽管希望与这一领域的其他世界专家建立联系，以加强所进行研究的结果和影响。

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.