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A142957
Duplicate of A038987.
0
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
1,1
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)
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.
EXAMPLE
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).
CROSSREFS
Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Sequence in context: A278209 A195732 A038987 * A191058 A080155 A235625
KEYWORD
dead
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
STATUS
approved