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A237606
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Positive integers k such that x^2 - 8xy + y^2 + k = 0 has integer solutions.
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11
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6, 11, 14, 15, 24, 35, 44, 51, 54, 56, 59, 60, 71, 86, 96, 99, 110, 119, 126, 131, 134, 135, 140, 150, 159, 176, 179, 191, 204, 206, 215, 216, 224, 231, 236, 239, 240, 251, 254, 275, 284, 294, 311, 315, 326, 335, 339, 344, 350, 359, 366, 371, 374, 375, 384
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OFFSET
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1,1
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COMMENTS
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Positive numbers of the form 15x^2 - y^2. The reduced form is -x^2 + 6xy + 6y^2.
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].
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].
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].
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.
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.
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.
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)
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LINKS
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EXAMPLE
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6 is in the sequence because x^2 - 8xy + y^2 + 6 = 0 has integer solutions, for example (x, y) = (1, 7).
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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