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

%I A001915 M3807 N1555 #56 Jul 24 2025 23:37:06
%S A001915 2,5,11,13,19,23,29,37,47,53,59,61,67,71,83,97,101,107,131,139,149,
%T A001915 163,167,173,179,181,191,193,197,211,227,239,263,269,293,307,311,313,
%U A001915 317,347,349,359,373,379,383,389,409,419,421,431,443,461,467,479,491,499,503,509,523
%N A001915 Primes p such that the congruence 2^x == 3 (mod p) is solvable.
%C A001915 The sequence is known to be infinite [Polya] - thanks to Pieter Moree and Daniel Stefankovic for this comment, Dec 21 2009.
%D A001915 M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 63.
%D A001915 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001915 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001915 Xianwen Wang, <a href="/A001915/b001915.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A001915 G. Polya, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002168812">Arithmetische Eigenschaften der Reihenentwicklungen rationaler Funktionen</a>, J. reine und angewandte Mathematik (Crelle), Volume 1921, Issue 151, Pages 1-31.
%p A001915 N:= 1000: # to search the first N primes
%p A001915 {2} union select(t -> numtheory[mlog](3,2,p) <> FAIL, {seq(ithprime(n),n=2..N)});
%p A001915 # _Robert Israel_, Feb 15 2013
%t A001915 Select[Prime[Range[120]], MemberQ[Table[Mod[2^x-3, #], {x, 0, #}], 0]&] (* _Jean-François Alcover_, Aug 29 2011 *)
%t A001915 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 *)
%o A001915 (PARI) isok(p) = isprime(p) && sum(k=0, (p-1), Mod(2, p)^k == 3); \\ _Michel Marcus_, Mar 12 2017
%o A001915 (PARI) is(n)=isprime(n) && (n==2 || #znlog(3, Mod(2, n))) \\ _Charles R Greathouse IV_, Aug 15 2018
%Y A001915 Cf. A001916, A050259, A123988.
%K A001915 nonn,easy,nice
%O A001915 1,1
%A A001915 _N. J. A. Sloane_
%E A001915 Better description from Joe K. Crump (joecr(AT)carolina.rr.com), Dec 11 2000
%E A001915 More terms from _David W. Wilson_, Dec 12 2000