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arXiv · 1711.05793

On the proximity of large primes

Minjia Shi·Florian Luca·Patrick Solé
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Abstract

By a sphere-packing argument, we show that there are infinitely many pairs of primes that are close to each other for some metrics on the integers. In particular, for any numeration basis $q$, we show that there are infinitely many pairs of primes the base $q$ expansion of which differ in at most two digits. Likewise, for any fixed integer $t,$ there are infinitely many pairs of primes, the first $t$ digits of which are the same. In another direction, we show that, there is a constant $c$ depending on $q$ such that for infinitely many integers $m$ there are at least $c\log \log m$ primes which differ from $m$ by at most one base $q$ digit.

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Minjia Shi, Florian Luca, Patrick Solé. 2017-11-15. On the proximity of large primes. https://arxiv.org/abs/1711.05793

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