Quadratic reciprocity
Fermat

Fermat proved (or claimed to have proved) a number of theorems about expressing a prime by a quadratic form:

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

Euler conjectured, and Lagrange proved, that

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