Papers › An algorithm and computation to verify Legendre's Conjecture up to 3.33·10¹³
An algorithm and computation to verify Legendre's Conjecture up to 3.33·10¹³
Jonathan Sorenson, Jonathan Webster
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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 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≤N in time O( N logN loglogN) and space N^(O(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 = 3.33·10¹³, 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.
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