A063980 Pillai primes: primes p such that there exists an integer m such that m! + 1 == 0 (mod p) and p != 1 (mod m).

23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, 227, 233, 239, 251, 257, 269, 271, 277, 293, 307, 311, 317, 359, 379, 383, 389, 397, 401, 419, 431, 449, 461, 463, 467, 479, 499, 503, 521, 557, 563, 569, 571, 577, 593, 599, 601, 607

Offset: 1

Author

R. K. Guy, Sep 08 2001

Keywords

• nonn
• nice

Comments

Hardy & Subbarao prove that this sequence is infinite. An upper bound can be extracted from their proof: a(n) < e^e^...^e^O(n log n) with e appearing n times. This tetrational bound could be improved with results on the disjointness of the factorizations of numbers of the form k! + 1. - Charles R Greathouse IV, Sep 15 2015

Named after the Indian mathematician Subbayya Sivasankaranarayana Pillai (1901-1950). - Amiram Eldar, Jun 16 2021

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly, Vol. 109, No. 6 (2002), pp. 554-559; alternative link.
• Wikipedia, Pillai prime.

Crossrefs

Smallest m is given in A063828, largest in A211411.

Programs

Mathematica

ok[p_] := (r = False; Do[If[Mod[m! + 1, p] == 0 && Mod[p, m] != 1, r = True; Break[]], {m, 2, p}]; r); Select[Prime /@ Range[111], ok] (* Jean-François Alcover, Apr 22 2011 *)
nn=1000; fact=1+Rest[FoldList[Times,1,Range[nn]]]; t={}; Do[p=Prime[i]; m=2; While[m
T. D. Noe, Apr 22 2011 *)

PARI

is(p)=my(t=Mod(5040,p)); for(m=8, p-2, t*=m; if(t==-1 && p%m!=1, return(isprime(p)))); 0 \\ Charles R Greathouse IV, Feb 10 2013

Extensions

More terms from David W. Wilson, Sep 08 2001