Abstract

Let a_{1}, …, a_{k} be positive integers generating the unit ideal, and j be a residue class modulo L = lcm(a_{1}, …, a_{k}). It is known that the function r(N) that counts solutions to the equation x_{1}a_{1}+…+x_{k}a_{k} = N in non-negative integers x_{i} is a polynomial when restricted to nonnegative integers N ≡ j (mod 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 𝔮 = gcd(a_{1}, a_{2})·gcd(a_{1}, a_{3})· gcd(a_{2}, a_{3}) of the L residue classes. The case 𝔮 = L plays a special role, and it is studied in more detail.