A107168 Primes of the form 4x^2 + 15y^2.

19, 31, 79, 139, 151, 199, 211, 271, 331, 379, 439, 499, 571, 619, 631, 691, 739, 751, 811, 859, 919, 991, 1039, 1051, 1171, 1231, 1279, 1291, 1399, 1459, 1471, 1531, 1579, 1699, 1759, 1831, 1879, 1951, 1999, 2011, 2131, 2179, 2239, 2251, 2311

Offset: 1

Author

T. D. Noe, May 13 2005

Keywords

• nonn
• easy

Comments

Discriminant = -240. See A107132 for more information.

Also, primes of form u^2+15v^2 where v is odd (and u is necessarily even), while A107152 (which can also be expressed as x^2+60y^2) has even v. The former is {19,31} mod 60 and the latter is {1,49} mod 60, but both can be reduced to the common congruence {1,19} mod 30. - Tito Piezas III, Jan 01 2009

Links

• Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Cf. A139827.

Programs

Magma

[p: p in PrimesUpTo(3000) | p mod 60 in [19, 31]]; // Vincenzo Librandi, Jul 25 2012

Mathematica

QuadPrimes2[4, 0, 15, 10000] (* see A106856 *)

PARI

list(lim)=my(v=List(),t); forprime(p=19,lim, t=p%60; if(t==19||t==31, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 10 2017

Formula

The primes are congruent to {19, 31} (mod 60). - T. D. Noe, May 02 2008