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%I #39 Sep 08 2022 08:44:53
%S 2,3,11,13,23,37,47,59,61,71,73,83,97,107,109,131,157,167,179,181,191,
%T 193,227,229,239,241,251,263,277,311,313,337,347,349,359,373,383,397,
%U 409,419,421,431,433,443,457,467,479,491,503,541,563,577,587,599,601
%N Primes p such that 3 is a square mod p.
%C Also primes congruent to {1, 2, 3, 11} mod 12.
%C The subsequence p = 1 (mod 4) corresponds to A068228 and only these entries of a(n) are squares mod 3 (from the quadratic reciprocity law). - _Lekraj Beedassy_, Jul 21 2004
%C Largest prime factors of n^2 - 3. - _Vladimir Joseph Stephan Orlovsky_, Aug 12 2009
%C Aside from 2 and 3, primes p such that Legendre(3, p) = 1. Bolker asserts there are infinitely many of these primes. - _Alonso del Arte_, Nov 25 2015
%C The associated bases of the squares are 1, 0, 5, 4, 7, 15, 12, 11, 8, 28, 21, 13...: 1^2 = 3 -1*2, 0^2 = 3-1*3, 5^2 = 3+ 2*11, 4^2 = 3+1*13, 7^2 = 3+2*23, 15^2 = 3+6*37, 12^2 = 3+3*47,... - _R. J. Mathar_, Feb 23 2017
%D Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 74, Theorem 25.3.
%H Vincenzo Librandi, <a href="/A038874/b038874.txt">Table of n, a(n) for n = 1..1000</a>
%H Tamara M. Lavshuk, <a href="https://mzsvfu.ru/index.php/mz/article/view/regular-polygons-and-polyhedra-over-finite-field">Regular polygons and polyhedra over finite field</a>, Mathematical Notes of NEFU, Vol 22 No 4 (2015). Mentions this sequence.
%F a(n) ~ 2n log n. - _Charles R Greathouse IV_, Nov 29 2016
%e 11 is in the sequence since the equation x^2 - 11y = 3 has solutions, such as x = 5, y = 2.
%e 13 is in the sequence since the equation x^2 - 13y = 3 has solutions, such as x = 4, y = 1.
%e 17 is not in the sequence because x^2 - 17y = 3 has no solutions in integers; Legendre(3, 17) = -1.
%p select(isprime, [2,3, seq(seq(6+s+12*i, s=[-5,5]),i=0..1000)]); # _Robert Israel_, Dec 23 2015
%t Select[Prime[Range[250]], MemberQ[{1, 2, 3, 11}, Mod[#, 12]] &] (* _Vincenzo Librandi_, Aug 08 2012 *)
%t Select[Flatten[Join[{2, 3}, Table[{12n - 1, 12n + 1}, {n, 50}]]], PrimeQ] (* _Alonso del Arte_, Nov 25 2015 *)
%o (Magma) [p: p in PrimesUpTo(1200) | p mod 12 in [1, 2, 3, 11]]; // _Vincenzo Librandi_, Aug 08 2012
%o (PARI) forprime(p=2, 1e3, if(issquare(Mod(3, p)), print1(p , ", "))) \\ _Altug Alkan_, Dec 04 2015
%Y Cf. A002313, A033203, A038873, A045331, A057125.
%Y If the first two terms are omitted we get A097933. A040101 is another sequence.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Henry Bottomley_, Aug 10 2000
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