Year: 2024
Type: article
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 $$n^2$$ n 2 and $$(n+1)^2$$ ( n + 1 ) 2 . Oppermann’s conjecture subsumes Legendre’s conjecture by claiming there are primes between $$n^2$$ n 2 and $$n(n+1)$$ n ( n + 1 ) and also between $$n(n+1)$$ n ( n + 1 ) and $$(n+1)^2$$ ( 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 $$n\le N$$ n ≤ N in time $$O( N \log N \log \log N)$$ O ( N log N log log N ) and space $$N^{O(1/\log \log N)}$$ 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 $$N = 2\cdot 10^{9}$$ N = 2 · 10 9 up to $$N = 7.05\cdot 10^{13} > 2^{46}$$ N = 7.05 · 10 13 > 2 46 , 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.
Source: Research in Number Theory
Authors: Jonathan Sorenson, Jonathan Webster
Institution: Butler University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,490
Funder: Butler University
OpenAlex ID: W4405103109