A Theorem of Legendre in $I\Delta_0+\Omega_1$
  We prove a classical theorem due to Legendre, about the existence of non trivial solutions of quadratic diophantine equations of the form $ax^2+by^2+cz^2=0$, in the weak fragment of Peano Arithmetic $I\Delta_0+\Omega_1$. 
Michele Bovenzi, Paola D'Aquino
math.LO