An algorithm and computation to verify Legendre’s conjecture up to 7·1013

Abstract

Listen

Translate article

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(NlogNloglogN) and space NO(1/loglogN). 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 2^{46}$$\\end{document}]]>, 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.