A038883 Odd primes p such that 13 is a square mod p.

3, 13, 17, 23, 29, 43, 53, 61, 79, 101, 103, 107, 113, 127, 131, 139, 157, 173, 179, 181, 191, 199, 211, 233, 251, 257, 263, 269, 277, 283, 311, 313, 337, 347, 367, 373, 389, 419, 433, 439, 443, 467, 491, 503, 521, 523, 547, 563, 569, 571, 599, 601, 607, 641

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Equivalently, by quadratic reciprocity (since 13 == 1 (mod 4)), primes p which are squares mod 13.

The squares mod 13 are 0, 1, 4, 9, 3, 12 and 10.

Also primes of the form x^2 + 3*x*y - y^2. Discriminant = 13. Class = 1. This was originally a separate entry, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 06 2008. R. J. Mathar proved that this coincides with the present sequence, Jul 22 2008

Primes p such that x^2 + x = 3 has a solution mod p (the solutions over the reals are (-1+-sqrt(13))/2). - Joerg Arndt, Jul 27 2011

Examples

			13 == 1 (mod 3) and 1 is a square, so 3 is on the list.
101 is prime and congruent to 7^2 = 49 == 10 (mod 13), so 101 is on the list.
		

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Peter Luschny, Binary Quadratic Forms
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
• D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Crossrefs

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (primes p such that d=13 is a square mod p). A038889 (d=17). A141111, A141112 (d=65).

Cf. A296937.

CF. A000040, A120330.

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Programs

Mathematica

Select[ Prime@ Range@ 118, JacobiSymbol[ #, 13] > -1 &] (* Robert G. Wilson v, May 16 2008 *)
Select[Flatten[Table[13n + {1, 3, 4, 9, 10, 12}, {n, 50}]], PrimeQ[#] &] (* Alonso del Arte, Sep 16 2012 *)

PARI

forprime(p=3,1e3,if(issquare(Mod(13,p)),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

PARI

select( {is_A038883(n)=bittest(5659,n%13)&&isprime(n)}, [0..666]) \\ M. F. Hasler, Feb 17 2022

Sage

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 3, -1])
print(Q.represented_positives(641, 'prime')) # Peter Luschny, Sep 20 2018

Formula

A000040 \ A120330 U {13}: Complement of A120330 in the primes, and 13. - M. F. Hasler, Feb 17 2022

Extensions

Edited by N. J. A. Sloane, Apr 27 2008, Jul 28 2008