A142957 Duplicate of A038987.

2, 3, 11, 31, 43, 47, 53, 61, 73, 79, 89, 97, 101, 103, 109, 113, 151, 163, 167, 191, 193, 197, 227, 229, 241, 269, 283, 293, 307, 313, 353, 379, 389, 397, 419, 421, 431, 449, 461, 463, 467, 479, 487, 491, 503, 509, 521, 547, 557, 571, 593, 607, 613, 617, 631

Offset: 1

Author

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (laucabfer(AT)alum.us.es), Jul 17 2008

Keywords

• dead

Comments

Previous name was: Primes of the form 3*x^2 + 5*x*y - 6*y^2 (as well as of the form 6*x^2 + 11*x*y + y^2).

Discriminant = 97. Class = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2-4ac.

Is this the same as A038987? - R. J. Mathar, Oct 12 2013

From Don Reble, Oct 06 2014: (Start)

G. B. Mathews ("Theory of Numbers" by Chelsea publishing) might have an answer to the relation with A038987. In point 59 on page 65 he claims that

- if X is a non-residue of a discriminant of a quadratic form, then X is not representable; and

- if X is a residue of D, then there is a quadratic form of determinant D which represents X.

If all forms of discriminant 97 are equivalent, then that might suffice. (Indeed, either +97 or -97 has class number 1; but I am not sure which sign matters, A003656 vs. A003173.)

(End)

From Jianing Song, Feb 24 2021: (Start)

Also primes of the form u^2 + u*v - 24*v^2. Substitute u, v by u = 9*x+22*y, v = 2*x+5*y gives 3*x^2 + 5*x*y - 6*y^2.

Yes, this is the same as A038987. For primes p being a (coprime) square modulo 97, they split in the ring Z[(1+sqrt(97))/2]. Since Z[(1+sqrt(97))/2] is a UFD, they are reducible in Z[(1+sqrt(97))/2], so we have p = e*(u + v*(1+sqrt(97))/2)*(u + v*(1-sqrt(97))/2) = e*(u^2 + u*v - 24*v^2), e = +-1. WLOG we can suppose e = 1, otherwise substitute u, v by 5035*u+27312*v and 1138*u+6173*v, then p = u^2 + u*v - 24*v^2. On the other hand, if p is a quadratic nonresidue modulo 97, then they remain inert in Z[(1+sqrt(97))/2] and hence cannot be represented as u^2 + u*v - 24*v^2. (End)

Examples

			a(6) = 47 because we can write 47 = 3*11^2 + 5*11*(-4) - 6*(-4)^2 (or 47 = 6*2^2 + 11*2*1 + 1^2).
		

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory.

Links

• 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

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).