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On positive integer solutions of the equation $xy+yz+xz=n$
Canadian mathematical bulletin
AS it had been recognized by Liouville, Hermite, Mordell and others, the number of non-negative integer solutions of the equation in the title is strongly related to the class number of quadratic forms with discriminant -n. The purpose of this note is to point out a deeper relation which makes it possible to derive a reasonable upper bound for the number of solutions. For a positive integer n let G(ri) denote the class number of binary quadratic forms aX 2 + 2bXY+ cY 2 with determinant b 2 -acdoi:10.4153/cmb-1996-024-5 fatcat:elm6tgphorgb7nte2lrkxnbuiq