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%I #27 Feb 19 2022 01:35:11
%S 2,3,11,31,43,47,53,61,73,79,89,97,101,103,109,113,151,163,167,191,
%T 193,197,227,229,241,269,283,293,307,313,353,379,389,397,419,421,431,
%U 449,461,463,467,479,487,491,503,509,521,547,557,571,593,607,613,617,631
%N Duplicate of A038987.
%C 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).
%C Discriminant = 97. Class = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2-4ac.
%C Is this the same as A038987? - _R. J. Mathar_, Oct 12 2013
%C From _Don Reble_, Oct 06 2014: (Start)
%C 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
%C - if X is a non-residue of a discriminant of a quadratic form, then X is not representable; and
%C - if X is a residue of D, then there is a quadratic form of determinant D which represents X.
%C 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.)
%C (End)
%C From _Jianing Song_, Feb 24 2021: (Start)
%C 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.
%C 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)
%D Z. I. Borevich and I. R. Shafarevich, Number Theory.
%H N. J. A. Sloane et al., <a href="/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a>: Index to related sequences, programs, references. OEIS wiki, June 2014.
%H D. B. Zagier, <a href="https://doi.org/10.1007/978-3-642-61829-1">Zetafunktionen und quadratische Körper</a>, Springer, 1981.
%e 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).
%Y Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
%K dead
%O 1,1
%A 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
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