A038874 Primes p such that 3 is a square mod p.

2, 3, 11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, 179, 181, 191, 193, 227, 229, 239, 241, 251, 263, 277, 311, 313, 337, 347, 349, 359, 373, 383, 397, 409, 419, 421, 431, 433, 443, 457, 467, 479, 491, 503, 541, 563, 577, 587, 599, 601

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also primes congruent to {1, 2, 3, 11} mod 12.

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

Largest prime factors of n^2 - 3. - Vladimir Joseph Stephan Orlovsky, Aug 12 2009

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

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

Examples

			11 is in the sequence since the equation x^2 - 11y = 3 has solutions, such as x = 5, y = 2.
13 is in the sequence since the equation x^2 - 13y = 3 has solutions, such as x = 4, y = 1.
17 is not in the sequence because x^2 - 17y = 3 has no solutions in integers; Legendre(3, 17) = -1.
		

References

• Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 74, Theorem 25.3.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Tamara M. Lavshuk, Regular polygons and polyhedra over finite field, Mathematical Notes of NEFU, Vol 22 No 4 (2015). Mentions this sequence.

Crossrefs

Cf. A002313, A033203, A038873, A045331, A057125.

If the first two terms are omitted we get A097933. A040101 is another sequence.

Programs

Magma

[p: p in PrimesUpTo(1200) | p mod 12 in [1, 2, 3, 11]]; // Vincenzo Librandi, Aug 08 2012

Maple

select(isprime, [2,3, seq(seq(6+s+12*i, s=[-5,5]),i=0..1000)]); # Robert Israel, Dec 23 2015

Mathematica

Select[Prime[Range[250]], MemberQ[{1, 2, 3, 11}, Mod[#, 12]] &] (* Vincenzo Librandi, Aug 08 2012 *)
Select[Flatten[Join[{2, 3}, Table[{12n - 1, 12n + 1}, {n, 50}]]], PrimeQ] (* Alonso del Arte, Nov 25 2015 *)

PARI

forprime(p=2, 1e3, if(issquare(Mod(3, p)), print1(p , ", "))) \\ Altug Alkan, Dec 04 2015

Formula

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 29 2016

Extensions

More terms from Henry Bottomley, Aug 10 2000