A139645 Primes of the form x^2 + 112*y^2.

113, 137, 193, 233, 281, 337, 401, 449, 457, 569, 617, 641, 673, 809, 953, 977, 1009, 1033, 1129, 1201, 1289, 1297, 1409, 1481, 1801, 1873, 1913, 2017, 2081, 2129, 2137, 2153, 2297, 2377, 2417, 2473, 2521, 2633, 2657, 2689, 2713, 2753, 2801

Offset: 1

Author

T. D. Noe, Apr 29 2008

Keywords

• nonn
• easy

Comments

Discriminant is -448. See A139643 for more information.

Primes of the form 8*n + 1 which cannot be expressed as 7*k - 1, 7*k - 2, or 7*k - 4. a(n)^3 == 1 (mod 56). - Gary Detlefs, Jan 26 2014

The primes are congruent to {1, 9, 25} (mod 56).

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 56 in {1, 9, 25}]; // Vincenzo Librandi, Jul 28 2012

Magma

k:=112; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

Maple

f:=n-> ceil((8*n+1)/7)-(8*n+1): for n from 1 to 350 do if isprime(8*n+1) and f(n)<>1 and f(n)<>2 and f(n)<>4 then print(8*n+1) fi od. # Gary Detlefs, Jan 26 2014

Mathematica

QuadPrimes2[1, 0, 112, 10000] (* see A106856 *)