A139492 Primes of the form x^2 + 5x*y + y^2 for x and y nonnegative.

7, 37, 43, 67, 79, 109, 127, 151, 163, 193, 211, 277, 331, 337, 373, 379, 421, 457, 463, 487, 499, 541, 547, 571, 613, 631, 673, 709, 739, 751, 757, 823, 877, 883, 907, 919, 967, 991, 1009, 1033, 1051, 1087, 1093, 1117, 1129, 1171, 1201, 1213, 1297, 1303

Offset: 1

Author

Artur Jasinski, Apr 24 2008

Keywords

• nonn

Comments

Reduced form is [1, 3, -3]. Discriminant = 21. Class number = 2.

Values of the quadratic form are {0, 1, 3, 4} mod 6, so this is a subsequence of A002476. - R. J. Mathar, Jul 30 2008

It can be checked that the primes p of the form x^2 + n*x*y + y^2, n >= 3, where x and y are nonnegative, depend on n mod 6 as follows: n mod 6 = 0 => p mod 12 = {1,5}; n mod 6 = 1 => p mod 12 = {1,7}; n mod 6 = 2 => p mod 12 = {1}; n mod 6 = 3 => p mod 12 = {1,5,7,11}; n mod 6 = 4 => p mod 12 = {1}; n mod 6 = 5 => p mod 12 = {1,7}. - Walter Kehowski, Jun 01 2008

Examples

			a(1) = 7 because we can write 7 = 1^2 + 5*1*1 + 1^2.
		

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory.
• David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Links

• Peter Luschny, Binary Quadratic Forms
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
• D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Crossrefs

Primes in A243172.

Cf. A002476, A007645, A007519, A033212, A033215, A068228, A107008, A107145, A107152, A139490, A139489, A139491, A139492, A141159.

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Programs

Mathematica

a = {}; w = 5; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a]

Sage

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 5, 1])
print(Q.represented_positives(1303, 'prime')) # Peter Luschny, May 12 2021