Separable Diophantine equations
Eric Temple Bell · Transactions of the American Mathematical Society · 1945
Theoretically, as noted by Skolem [3, p. 21(1), the general problem of algebraic diophantine analysis is reducible to the case in which occur only equations and inequalities of degree not higher than the second. For the extensive class of separable systems defined in ?6, this reduction can be performed effectively, eventuating in the complete integer solutions of the equations concerned. The general method is strictly elementary, but none the less powerful within its natural range on that account. Among the more immediate applications are complete solutions of certain types of homogeneous equations of the second degree, only special cases of which have been solved hitherto by advanced methods, including that of generalized quaternions. The equation x:+ +x2=y2, for example, does not seem to be adapted to such methods, as a sum of n squares is not factorable, for all n, in a ring. For simplicity of statement only, the method is presented for the domain of rational integers. A few slight and obvious verbal changes suffice to extend the entire discussion to any unique factorization domain, in particular to domains in which there is a Euclidean algorithm. The extension to principal ideal rings of algebraic integers, for instance, yields results of interest in the domain of rational integers.