A001915 Primes p such that the congruence 2^x == 3 (mod p) is solvable.

2, 5, 11, 13, 19, 23, 29, 37, 47, 53, 59, 61, 67, 71, 83, 97, 101, 107, 131, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 211, 227, 239, 263, 269, 293, 307, 311, 313, 317, 347, 349, 359, 373, 379, 383, 389, 409, 419, 421, 431, 443, 461, 467, 479, 491, 499, 503, 509, 523

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

The sequence is known to be infinite [Polya] - thanks to Pieter Moree and Daniel Stefankovic for this comment, Dec 21 2009.

References

• M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 63.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Xianwen Wang, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• G. Polya, Arithmetische Eigenschaften der Reihenentwicklungen rationaler Funktionen, J. reine und angewandte Mathematik (Crelle), Volume 1921, Issue 151, Pages 1-31.

Crossrefs

Cf. A001916, A050259, A123988.

Programs

Maple

N:= 1000: # to search the first N primes
{2} union select(t -> numtheory[mlog](3,2,p) <> FAIL, {seq(ithprime(n),n=2..N)});
# Robert Israel, Feb 15 2013

Mathematica

Select[Prime[Range[120]], MemberQ[Table[Mod[2^x-3, #], {x, 0, #}], 0]&] (* Jean-François Alcover, Aug 29 2011 *)
Monitor[aaa=Reap[Do[p=Prime[m];sol=MultiplicativeOrder[2,p,{3}];If[IntegerQ[sol],Sow[p]],{m,1000}]],{m}];tmp=Transpose[{1+Range[Length[aaa[[2,1]]]],aaa[[2,1]]}] (* Xianwen Wang, Jul 22 2025 *)

PARI

isok(p) = isprime(p) && sum(k=0, (p-1), Mod(2, p)^k == 3); \\ Michel Marcus, Mar 12 2017

PARI

is(n)=isprime(n) && (n==2 || #znlog(3, Mod(2, n))) \\ Charles R Greathouse IV, Aug 15 2018

Extensions

Better description from Joe K. Crump (joecr(AT)carolina.rr.com), Dec 11 2000

More terms from David W. Wilson, Dec 12 2000