A Unified Approach to Calculating Sylvester Sums

Abstract: In the context of the Frobenius coin problem, given two relatively prime positive integers $a$ and $b$, the set of nonrepresentable numbers consists of positive integers that cannot be expressed as nonnegative integer combination of $a$ and $b$. This work provides a formula for calculating the power sums of all nonrepresentable numbers, also known as the Sylvester sums. Although alternative formulas exist in the literature, our approach is based on an elementary observation. We consider the set of natural numbers from $1$ to $ab - 1$ and compute their total sum in two distinct ways, which leads naturally to the desired Sylvester sums. This method connects an analytic identity with a combinatorial viewpoint, giving a new way to understand these classical quantities. Furthermore, in this paper, we establish a criterion using the division algorithm to determine whether a given positive integer is nonrepresentable.