Editing Quadratic reciprocity (section)

===Fermat===
Fermat proved<ref>Lemmermeyer, pp. 2&ndash;3</ref> (or claimed to have proved)<ref>Gauss, DA, art. 182</ref> a number of theorems about expressing a prime by a quadratic form:

:<math>\begin{align}
p=x^2+ y^2 \qquad &\Longleftrightarrow \qquad p=2 \quad \text{ or } \quad p\equiv 1 \bmod{4} \\
p=x^2+2y^2 \qquad &\Longleftrightarrow \qquad p=2 \quad \text{ or } \quad p\equiv 1, 3 \bmod{8} \\
p=x^2+3y^2 \qquad &\Longleftrightarrow \qquad p=3 \quad \text{ or } \quad p\equiv 1 \bmod{3} \\
\end{align}</math>

He did not state the law of quadratic reciprocity, although the cases −1, ±2, and ±3 are easy deductions from these and others of his theorems.

He also claimed to have a proof that if the prime number ''p'' ends with 7, (in base 10) and the prime number ''q'' ends in 3, and ''p'' ≡ ''q'' ≡ 3 (mod 4), then

:<math>pq=x^2+5y^2.</math>

Euler conjectured, and Lagrange proved, that<ref>Lemmermeyer, p. 3</ref>

:<math>\begin{align}
p &\equiv 1, 9 \bmod{20}\quad \Longrightarrow \quad p = x^2+5y^2 \\
p, q &\equiv 3, 7 \bmod{20} \quad \Longrightarrow \quad pq=x^2+5y^2
\end{align}</math>

Proving these and other statements of Fermat was one of the things that led mathematicians to the reciprocity theorem.