An elementary remark on the distribution of integers representable by a positive-definite integral binary quadratic form (Q1842014)

Let \(f(X,Y)= aX^ 2+ bXY+ cY^ 2\) be a positive-definite integral binary quadratic form of discriminant \(-\Delta= b^ 2- 4ac<0\). Let \(m_ 1\) be the least positive integer represented by \(f\). Then, for every integer \(n\geq m_ 1\), there exist integers \(x\) and \(y\) such that \[ n< ax^ 2+ bxy+ cy^ 2< n+ 2m_ 1^{1/4} \Delta^{1/4} n^{1/4}+ m_ 1. \] This is proved in an elementary way. For \(f(X, Y)= aX^ 2+ cY^ 2\) and \(n\geq {{c^ 3}\over {a^ 2}}\) this is further improved, generalizing a theorem of \textit{S. Uchiyama} [J. Fac. Sci., Hokkaido Univ., I. Ser. 18, 124-127 (1964; Zbl 0199.087)].