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In a recent work, the present author generalized a fundamental result ofGauss related to quadratic reciprocity, and also showed that the above resultof Gauss is equivalent to a special case of a well-known result of Sylvesterrelated to the Frobenius coin problem. In this note, we use this equivalence toshow that the above generalization of the result of Gauss naturally leads to aninteresting generalization of the result of Sylvester. To be precise, for givenpositive coprime integers $a$ and $b$, and for a family of values of $k$ in theinterval $0 \leq k < (a-1)(b-1)$, 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 Jacobisymbols using floor function sums. Our proof provides a natural straightforwardgeneralization of the Gauss-Eisenstein proof of the law of quadraticreciprocity for Jacobi symbols.

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