

A038874


Primes p such that 3 is a square mod p.


11



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
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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
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 = 31*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



FORMULA



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

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


KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



