Abstract:

Let $a_1,\ldots,a_k$ be positive integers generating the unit ideal, and $j$ be a residue class modulo $L = \operatorname{lcm}(a_1,\ldots,a_k)$. It is known that the function $r(N)$ that counts solutions to the equation $x_1a_1 + \ldots + x_ka_k = N$ in non-negative integers $x_i$ is a polynomial when restricted to non-negative integers $N \equiv j \pmod L$. Here we give, in the case of $k=3$, exact formulas for these polynomials up to the constant terms, and exact formulas including the constants for $\mathfrak{q}= \gcd(a_1,a_2) \cdot \gcd(a_1,a_3) \cdot \gcd(a_2,a_3)$ of the $L$ residue classes. The case $\mathfrak{q}= L$ plays a special role, and it is studied in more detail.