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

1,1

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

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.

Formula

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

Example

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.

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 *)

Prog

(Magma) [p: p in PrimesUpTo(1200) | p mod 12 in [1, 2, 3, 11]]; // Vincenzo Librandi, Aug 08 2012
(PARI) forprime(p=2, 1e3, if(issquare(Mod(3, p)), print1(p , ", "))) \\ Altug Alkan, Dec 04 2015

Crossrefs

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

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

Sequence in context: A090707 A062350 A163498 * A164624 A215819 A040101

Adjacent sequences: A038871 A038872 A038873 * A038875 A038876 A038877

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Henry Bottomley, Aug 10 2000

Status

approved