Small solutions of the Legendre equation
Let \(a\), \(b\), \(c\) be positive integers. The Legendre equation \(ax^2+ by^2- cz^2= 0\) is said to be in normal form if \(a\), \(b\), \(c\) are squarefree and pairwise coprime. If the Legendre equation \(ax^2+ by^2- cz^2= 0\) is in normal form and has a solution in integers \(x\), \(y\), \(z\) not all zero, then there is a nontrivial solution with \(| x|\leq \sqrt{bc}\), \(| y|\leq \sqrt{ac}\) and \(| z|\leq \sqrt{ab}\). This result was first proved by \textit{L. Holzer} [Can. J. Math. 2, 238-244 (1950; Zbl 0037.02602)] using a deep result. The first elementary proof was given by \textit{L. J. Mordell} [J. Number Theory 1, 1-3 (1969; Zbl 0169.06104)] with a small gap filled by \textit{K. S. Williams} [Util. Math. 34, 65-72 (1988; Zbl 0666.10010)]. In this paper, the authors give a new elementary proof of this theorem by showing that the lattice of solutions of the congruence \[ ax^2+ by^2- cz^2\equiv 0\pmod{abc}, \] which is constructed in most textbook proofs of Legendre's necessary and sufficient condition for the solvability of \(ax^2+ by^2- cz^2\equiv 0\), always contains a solution satisfying the above bounds.
scientific article; zbMATH DE number 1207965