A001915 - OEIS

%I M3807 N1555 #43 Feb 20 2021 00:37:44

%S 2,5,11,13,19,23,29,37,47,53,59,61,67,71,83,97,101,107,131,139,149,

%T 163,167,173,179,181,191,193,197,211,227,239,263,269,293,307,311,313,

%U 317,347,349,359,373,379,383,389,409,419,421,431,443,461,467,479,491,499,503,509,523

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

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

%D M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 63.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001915/b001915.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 N:= 1000: # to search the first N primes

%p {2} union select(t -> numtheory[mlog](3,2,p) <> FAIL, {seq(ithprime(n),n=2..N)});

%p # _Robert Israel_, Feb 15 2013

%t Select[Prime[Range[120]], MemberQ[Table[Mod[2^x-3, #], {x, 0, #}], 0]&] (* _Jean-François Alcover_, Aug 29 2011 *)

%o (PARI) isok(p) = isprime(p) && sum(k=0, (p-1), Mod(2, p)^k == 3); \\ _Michel Marcus_, Mar 12 2017

%o (PARI) is(n)=isprime(n) && (n==2 || #znlog(3, Mod(2, n))) \\ _Charles R Greathouse IV_, Aug 15 2018

%Y Cf. A001916, A050259, A123988.

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_

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

%E More terms from _David W. Wilson_, Dec 12 2000