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Journal Article

On the proximity of large primes

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Luca,  Florian
Max Planck Institute for Mathematics, Max Planck Society;

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arXiv:1711.05793.pdf
(Preprint), 125KB

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Citation

Shi, M., Luca, F., & Solé, P. (2018). On the proximity of large primes. Mathematica Slovaca, 68(5), 981-986. doi:10.1515/ms-2017-0160.


Cite as: https://hdl.handle.net/21.11116/0000-0003-DE28-7
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.