On positive integer solutions of the equation $xy+yz+xz=n$

Al-Zaid Hassan, B. Brindza, Á. Pintér
1996 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 -ac
more » ... -n. Generalizing some earlier results, Mordell ([Ml], [M2]) proved that the number of non-negative integer solutions of the equation (1) xy+yz + xz = n is 3G(n) if a weight is attached to a solution with xyz = 0. His argument is based upon a one-to-one correspondence between the reduced quadratic forms AX 2 + 2BXY+CY 2 , and the non-negative solutions x, y, z of (1) given by A = x + y, \B\ = x, C = x + z. However, the counting of strictly positive integer solutions seems to be a different and harder problem. It was verified ([K]) that the equation (1) (in positive integers) has solution for all n < 10 7 except the numbers n = 1, 2, 4, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330 and 462 which is the biggest one. Let h(D) and h(D) denote the ideal class number of the field Q(y/-D) and the class number of the forms aX 2 + bXY+cY 2 with discriminant b 2 -4ac = -D, respectively. In our equation n is not necessarily square-free and it does not satisfy certain prescribed congruences modulo 4, thus the relation between the class numbers h(D), h(D), and the number of solutions of ( 1 ) is not that straightforward, apart from the simple inequality max{h(D\ h(D)} < G(D). Let S(n) denote the number of integer solutions of (1) with 0 < z < y < x and e be a positive number. Then we have
doi:10.4153/cmb-1996-024-5 fatcat:elm6tgphorgb7nte2lrkxnbuiq