A Theorem of Legendre in $I\Delta_0+\Omega_1$

TL;DR

This paper proves a classical number theory theorem within a weak arithmetic system, demonstrating the existence of solutions to certain quadratic equations in a limited logical framework.

Contribution

It establishes Legendre's theorem in the weak fragment of Peano Arithmetic, expanding the understanding of number theory in foundational logical systems.

Findings

Legendre's theorem holds in $I riangle_0+ ext{ extOmega}_1$

Existence of solutions to quadratic equations proven in weak arithmetic

Bridges classical number theory with weak logical systems

Abstract

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 $I\Delta_0+\Omega_1$.