Generalization of a Result of Sylvester Regarding the Frobenius Coin Problem and an Elementary Proof of Eisenstein's Lemma for Jacobi Symbols

TL;DR

This paper extends Sylvester's Frobenius coin problem result using a generalized Gauss result, introduces a new formula for counting representable integers, and offers an elementary proof of Eisenstein's Lemma for Jacobi symbols.

Contribution

It generalizes Sylvester's Frobenius coin problem result and provides an elementary proof of Eisenstein's Lemma for Jacobi symbols, connecting quadratic reciprocity with coin problems.

Findings

Derived a formula for counting integers representable as linear combinations of two coprime numbers.

Established a natural generalization of the Gauss-Eisenstein proof for quadratic reciprocity.

Provided an elementary proof of Eisenstein's Lemma for Jacobi symbols.

Abstract

In a recent work, the present author generalized a fundamental result of Gauss related to quadratic reciprocity, and also showed that the above result of Gauss is equivalent to a special case of a well-known result of Sylvester related to the Frobenius coin problem. In this note, we use this equivalence to show that the above generalization of the result of Gauss naturally leads to an interesting generalization of the result of Sylvester. To be precise, for given positive coprime integers $a$ and $b$, and for a family of values of $k$ in the interval $0 \leq k < (a-1)(b-1)$, we find the number of nonnegative integers $\leq k$ that can be expressed in the form $ax+by$ for nonnegative integers $x$ and $y$. We also give an elementary proof of Eisenstein's Lemma for Jacobi symbols using floor function sums. Our proof provides a natural straightforward generalization of the Gauss-Eisenstein proof of the law of quadratic reciprocity for Jacobi symbols.