README.md

LegendreQF

Lean code formalizing a proof of Legendre's theorem on diagonal ternary quadratic forms

This repository contains lean3 code (using mathlib3; in the lean3 directory) and lean4 code (using mathlib4; in the lean4 directory) formalizing a proof of Legendre's Theorem on diagonal ternary quadratic forms:

Theorem. Let $a, b, c$ be squarefree integers that are coprime in pairs. Then the equation $$a x^2 + b y^2 + c z^2 = 0$$ has a nontrivial (i.e., not all values are zero) solution in integers if and only if

1. $a, b, c$ do not have the same sign,
2. $-ab$ is a square mod $c$,
3. $-bc$ is a square mod $a$,
4. $-ca$ is a sqaure mod $b$.

It is easy to see that the conditions are necessary; the main part of the statement is that they are also sufficient.

The statement implies the Hasse Principle (for the ternary quadratic forms considered): A ternary quadratic form as above has a nontrivial rational zero if and only if it has a nontrivial real zero and nontrivial $p$-adic zeros for all primes $p$. (However, this is not (yet) part of the formalization.)