Legendre-Teege Reciprocity

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.