A033221 Primes of form x^2+31*y^2.

31, 47, 67, 131, 149, 173, 227, 283, 293, 349, 379, 431, 521, 577, 607, 617, 653, 811, 839, 853, 857, 919, 937, 971, 1031, 1063, 1117, 1187, 1213, 1237, 1259, 1303, 1327, 1451, 1493, 1523, 1559, 1583, 1619, 1663, 1721, 1723, 1741, 1879, 1931, 1973, 1993, 2003, 2017, 2153, 2273, 2333, 2341, 2521, 2531, 2539, 2543, 2609, 2707, 2711, 2713, 2767, 2797

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also primes of the form x^2+xy+8y^2. - N. J. A. Sloane, Jun 02 2014

Also primes of the form x^2-xy+8y^2 with x and y nonnegative. - T. D. Noe, May 07 2005

Primes p such that the polynomial X^3 + X + 1 splits mod p (see Williams and Hudson link). - Robert Israel, Jun 01 2020

References

• David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Links

• Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 2000 terms from Vincenzo Librandi]
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
• K. Williams and R. Hudson, Representation of primes by the principal form of discriminant -D when the classnumber h(-D) is 3, Acta Arithmetica 57.2 (1991): 131-153.

Crossrefs

Primes in A243176.

Programs

Maple

N:= 10000: # for terms <= N
S:= select(isprime,{31,seq(seq(x^2+31*y^2, y=1..floor(sqrt((N-x^2)/31))),
  x=1..floor(sqrt(N)))}):
sort(convert(S,list)); # Robert Israel, Jun 01 2020

Mathematica

QuadPrimes2[1, 0, 31, 10000] (* see A106856 *)