##
**
Generalization of a Result of Sylvester Related to the Frobenius Coin Problem
**

###
Damanvir Singh Binner

Department of Mathematics

Simon Fraser University

Burnaby, BC V5A 1S6

Canada

**Abstract:**

In a recent paper, 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 ≤ *k* < (*a* – 1)(*b* – 1),
we find the number
of nonnegative integers ≤ *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.

**
Full version: pdf,
dvi,
ps,
latex
**

Received July 10 2021; revised versions received July 20 2021; July 23 2021; August 6 2021; August 10 2021.
Published in *Journal of Integer Sequences*,
August 17 2021.

Return to
**Journal of Integer Sequences home page**