Quadratic diophantine equations

Abstract Tartakowsky (1929) proved that a positive definite quadratic form, with integral coefficients, in 5 or more variables represents all but at most finitely many of the positive integers not excluded by congruence considerations. Tartakowsky’s argument does not lead to any estimate for a positive integer which, though not so excluded, is not represented by the quadratic form. Here estimates for such an integer are obtained, in terms of the coefficients of the quadratic form. To simplify the argument and improve the estimates, the problem is slightly generalized (by considering a Diophantine equation with linear terms). A combination of analytical and arithmetical methods is needed.