Document Type

Article

Publication Date

2025

Publication Title

Research in Number Theory

First Page

3

Last Page

9

DOI

https://doi.org/10.1007/s40993-024-00589-4

Additional Publication URL

https://link.springer.com/article/10.1007/s40993-024-00589-4#rightslink

Abstract

We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre’s conjecture claims that for every positive integer n, there exists a prime between n2 and (n + 1)2. Oppermann’s conjecture subsumes Legendre’s conjecture by claiming there are primes between n2 and n(n + 1) and also between n(n + 1) and (n + 1)2. Using Cramér’s conjecture as the basis for a heuristic run-time analysis, we show that our algorithm can verify Oppermann’s conjecture, and hence also Legendre’s conjecture, for all n ≤ N in time O(N log N log log N) and space NO(1/ log log N) . We implemented a parallel version of our algorithm and improved the empirical verification of Oppermann’s conjecture from the previous N = 2 · 109 up to N = 7.05 · 1013 > 246, so we were finding 27 digit primes. The computation ran for about half a year on each of two platforms: four Intel Xeon Phi 7210 processors using a total of 256 cores, and a 192-core cluster of Intel Xeon E5-2630 2.3GHz processors.

Rights

This article was originally published in Research in Number Theory, 2025, Volume 11.

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