A Theorem of Legendre in $I{\Delta }_0+{\Omega }_1$

Abstract

We prove a classical theorem due to Legendre, about the existence of non trivial solutions of quadratic diophantine equations of the form $a{x}^2+b{y}^2+c{z}^2=0$, in the weak fragment of Peano Arithmetic $I{\Delta }_0+{\Omega }_1$.