Small Solutions of the Legendre Equation Original Research Article

Holzer, using a deep theorem, proved that if the Legendre equationax2+by2−cz2=0 is in normal form and has a nonzero solution then there is a solution withimageandimage. The first elementary proof of this result was given by Mordell with a small gap filled by Williams. We give a new elementary proof of this theorem. We show in fact that the lattice of solutions of the congruenceax2+by2−cz2≡0 (mod abc) that is constructed in most textbook proofs of Legendreʹs theorem always contains such a solution.