cp's OEIS Frontend

A237606 Positive integers k such that x^2 - 8xy + y^2 + k = 0 has integer solutions.

%I A237606 #25 Dec 26 2024 08:25:08
%S A237606 6,11,14,15,24,35,44,51,54,56,59,60,71,86,96,99,110,119,126,131,134,
%T A237606 135,140,150,159,176,179,191,204,206,215,216,224,231,236,239,240,251,
%U A237606 254,275,284,294,311,315,326,335,339,344,350,359,366,371,374,375,384
%N A237606 Positive integers k such that x^2 - 8xy + y^2 + k = 0 has integer solutions.
%C A237606 From _Klaus Purath_, Feb 17 2024: (Start)
%C A237606 Positive numbers of the form 15x^2 - y^2. The reduced form is -x^2 + 6xy + 6y^2.
%C A237606 Even powers of terms as well as products of an even number of terms belong to A243188. This can be proved with respect to the forms [a,0,-c] and [a, 0, +c] by the following identities: (au^2 - cv^2)(ax^2 - cy^2) = (aux + cvy)^2 - ac(uy + vx)^2 and (au^2 + cv^2)(ax^2 + cy^2) = (aux - cvy)^2 + ac(uy + vx)^2 for all a, c, u, v, x, y in R. This can be verified by expanding both sides of the equations. Generalization (conjecture): This multiplication rule applies to all sequences represented by any binary quadratic form [a, b, c].
%C A237606 Odd powers of terms as well as products of an odd number of terms belong to the sequence. This can be proved with respect to the forms [a,0,-c] and [a, 0, +c] by the following identities: (as^2 - ct^2)(au^2 - cv^2)(ax^2 - cy^2) = a[s(aux + cvy) + ct(uy + vx)]^2 - c[as(uy + vx) + t(aux + cvy)]^2 and (as^2 + ct^2)(au^2 + cv^2)(ax^2 + cy^2) = a[s(aux - cvy) - ct(uy + vx)]^2 + c[as(uy + vx) + t(aux - cvy)]^2 for all a, c, s, t, u, v, x, y in R. This can be verified by expanding both sides of the equations. Generalization (conjecture): This multiplication rule applies to all sequences represented by any binary quadratic form [a, b, c].
%C A237606 If we denote any term of this sequence by B and correspondingly of A243189 by C and of A243190 by D, then B*C = D, C*D = B and B*D = C. This can be proved by the following identities, where the sequence (B) is represented by [kn, 0, -1], (C) by [n, 0, -k] and (D) by [k, 0, -n].
%C A237606 Proof of B*C = D: (knu^2 - v^2)(nx^2 - ky^2) =  k(nux + vy)^2 - n(kuy + vx)^2 for k, n, u, v, x, y in R.
%C A237606 Proof of C*D = B: (nu^2 - kv^2)(kx^2 - ny^2) = kn(ux + vy)^2 - (nuy + kvx)^2 for k, n, u, v, x, y in R.
%C A237606 Proof of B*D = C: (knu^2 - v^2)(kx^2 - ny^2) = n(kux + vy)^2 - k(nuy + vx)^2 for k, n, u, v, x, y in R. This can be verified by expanding both sides of the equations.
%C A237606 Generalization (conjecture): If there are three sequences of a given positive discriminant that are represented by the forms [a1, b1, c1], [a2, b2, c2] and [a1*a2, b3, c3] for a1, a2 != 1, then the BCD rules apply to these sequences. (End)
%H A237606 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%e A237606 6 is in the sequence because x^2 - 8xy + y^2 + 6 = 0 has integer solutions, for example (x, y) = (1, 7).
%Y A237606 Cf. A070997 (k = 6), A199336 (k = 14), A001091 (k = 15), A077248 (k = 35).
%Y A237606 Cf. A031363, A084917, A237351, A237599, A237609, A237610.
%Y A237606 For primes see A141302.
%Y A237606 Cf. A378710, A378711 (subsequence of properly represented numbers and fundamental solutions).
%K A237606 nonn
%O A237606 1,1
%A A237606 _Colin Barker_, Feb 10 2014