Progress Report on a Wilson Prime Search

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Progress report on a Wilson prime search David Harvey (with Edgar Costa and Robert Gerbicz) University of New South Wales 10th July 2012, ANTS X, UC San Diego David Harvey (with Edgar Costa and Robert Gerbicz) Progress report on a Wilson prime search Wilson primes For prime p, deﬁne the Wilson quotient (p − 1)! + 1 w = : p p Wilson's theorem: wp 2 Z: A Wilson prime is a prime with wp = 0 (mod p). Equival ly (p − 1)! = −1 (mod p2): David Harvey (with Edgar Costa and Robert Gerbicz) Progress report on a Wilson prime search Previous work Only known Wilson primes: 5, 13, 563. Crandall{Dilcher{Pomerance (1997): no others for p < 500 000 000: Carlisle{Crandall{Rodenkirch (2008, unpublished): p < 6 000 000 000: Heuristically X 1 #fWilson primes p < xg ∼ ∼ log log x: p p<x This does go to inﬁnity... but very slowly. David Harvey (with Edgar Costa and Robert Gerbicz) Progress report on a Wilson prime search Algorithms for computing wp I Naive: O(p1+o(1)) bit operations. I Baby-step/giant-step (Strassen): O(p1=2+o(1)) bit operations. I New algorithm: compute wp for all p < N using only O((log p)4+o(1)) bit operations per prime on average. Polynomial time! David Harvey (with Edgar Costa and Robert Gerbicz) Progress report on a Wilson prime search Current computation At this moment running on 500{1000 cores at NYU & UNSW. We have checked all p < 1 000 000 000 000 and larger p in some intervals. Goal: p < 10 000 000 000 000: About 8% chance of success for 1012 < p < 1013. p wp 11 774 118 061 −1 (largest known p with jwpj = 1) 1 239 053 554 603 −4 (smallest known nonzero jwp=pj) Table: Some close calls David Harvey (with Edgar Costa and Robert Gerbicz) Progress report on a Wilson prime search Another announcement Our new exascale machine arrived about 7 weeks ago: Jesse Oliver Harvey David Harvey (with Edgar Costa and Robert Gerbicz) Progress report on a Wilson prime search.

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