A139861 Primes of the form 2x^2 + 65y^2.

2, 67, 73, 83, 97, 137, 163, 193, 227, 307, 353, 457, 577, 587, 593, 617, 643, 683, 787, 827, 947, 977, 1033, 1097, 1123, 1163, 1217, 1307, 1523, 1553, 1627, 1657, 1697, 1723, 1747, 1753, 1787, 1867, 1913, 1987, 2017, 2113, 2137, 2153, 2203

Offset: 1

Author

T. D. Noe, May 02 2008

Keywords

• nonn
• easy

Comments

Discriminant = -520. See A139827 for more information.

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)

Programs

Magma

[ p: p in PrimesUpTo(3000) | p mod 520 in {2, 33, 57, 67, 73, 83, 97, 123, 137, 163, 177, 187, 193, 203, 227, 267, 297, 307, 323, 353, 427, 457, 473, 483, 513}]; // Vincenzo Librandi, Jul 29 2012

Mathematica

QuadPrimes2[2, 0, 65, 10000] (* see A106856 *)

PARI

list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\2), w=2*x^2; for(y=0, sqrtint((lim-w)\65), if(isprime(t=w+65*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Mar 07 2017

Formula

The primes are congruent to {2, 33, 57, 67, 73, 83, 97, 123, 137, 163, 177, 187, 193, 203, 227, 267, 297, 307, 323, 353, 427, 457, 473, 483, 513} (mod 520).