A139874 Primes of the form 3x^2 + 56y^2.

3, 59, 83, 131, 227, 251, 419, 467, 563, 587, 971, 1091, 1259, 1307, 1427, 1571, 1811, 1907, 1931, 1979, 2099, 2243, 2267, 2411, 2579, 2819, 2939, 3083, 3251, 3323, 3491, 3659, 3779, 3923, 3947, 4091, 4259, 4283, 4451, 4787, 4931, 5003

Offset: 1

Author

T. D. Noe, May 02 2008

Keywords

• nonn
• easy

Comments

Discriminant = -672. See A139827 for more information.

Except for 3, also primes of the form 20x^2 + 12xy + 27y^2. See A140633. - T. D. Noe, May 19 2008

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

[3] cat [ p: p in PrimesUpTo(6000) | p mod 168 in {59, 83, 131}]; // Vincenzo Librandi, Jul 30 2012

Mathematica

QuadPrimes2[3, 0, 56, 10000] (* see A106856 *)

PARI

list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\3), w=3*x^2; for(y=0, sqrtint((lim-w)\56), if(isprime(t=w+56*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Mar 07 2017

Formula

Except for 3, the primes are congruent to {59, 83, 131} (mod 168).