Legendre-Teege Reciprocity

TL;DR

This paper provides a detailed exposition of the historical proofs of the law of Quadratic Reciprocity, including Legendre's initial attempt, Teege's rigorous proof, and Rogers' implicit proof, which were previously not thoroughly documented.

Contribution

It offers the first comprehensive and detailed exposition of the proofs by Legendre, Teege, and Rogers, clarifying their contributions to quadratic reciprocity.

Findings

Clarifies the historical development of quadratic reciprocity proofs

Provides detailed explanations of Teege's and Rogers' proofs

Fills a gap in mathematical literature on quadratic reciprocity proofs

Abstract

Legendre published the first attempted proof of the law of Quadratic Reciprocity. But, in its final form (1797), it had a gap. Some 125 years later Herman Teege published the first rigorous proof of the unproven hypothesis which formed that gap. Then, 48 years later Kenneth Rogers published a second (but implicit) proof. These proofs lifted Legendre's attempt to the list of complete proofs. No detailed exposition of these proofs appears in the literature. Our paper fills that gap.