An algorithm and computation to verify Legendre's Conjecture up to $3.33\cdot10^{13}$

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 $n^2$ and $(n+1)^2$. Oppermann's conjecture subsumes Legendre's conjecture by claiming there are primes between $n^2$ and $n(n+1)$ and also between $n(n+1)$ and $(n+1)^2$. Using Cram\'er'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 $n\le N$ in time $O( N \log N \log \log N)$ and space $N^{O(1/\log \log N)}$. We implemented a parallel version of our algorithm and improved the empirical verification of Oppermann's conjecture from the previous $N = 2\cdot 10^{9}$ up to $N = 3.33\cdot 10^{13}$, so we were finding $27$ digit primes. The computation ran for about half a year on four Intel Xeon Phi $7210$ processors using a total of $256$ cores.