Gauss's Lemma, Eisenstein's Lemma and a new formula for Jacobi Symbols

Damanvir Singh Binner

Published 2021-02-16Version 1

In a recent work, we proved that a fundamental result of Gauss related to quadratic reciprocity is equivalent to a special case of a well known result of Sylvester related to the Frobenius coin problem. In this note, we appropriately generalize Gauss's result so that it becomes equivalent to the general version of Sylvester's result. This generalization of Gauss's result naturally leads us to another proof of Gauss's Lemma and Eisenstein's Lemma for Jacobi symbols. Using our work, we also obtain a new formula for the Jacobi symbols.