On the Number of Representations of an Integer by a Linear Form

Let $a_1,\ldots,a_k$ be positive integers generating the unit ideal, and

be a residue class modulo $L = \operatorname{lcm}(a_1,\ldots,a_k)$ . It is known that the function

that counts solutions to the equation $x_1a_1 + \ldots + x_ka_k = N$ in non-negative integers

is a polynomial when restricted to non-negative integers $N \equiv j \pmod L$ . Here we give, in the case of

, 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

residue classes. The case $\mathfrak{q}= L$ plays a special role, and it is studied in more detail.

Received June 23 2005; revised version received October 19 2005. Published in Journal of Integer Sequences October 20 2005.

On the Number of Representations of an Integer by a Linear Form

Gil Alon and Pete L. Clark 1126 Burnside Hall Department of Mathematics and Statistics McGill University 805 Sherbrooke West Montreal, QC H3A 2K6 Canada

Gil Alon and Pete L. Clark
1126 Burnside Hall
Department of Mathematics and Statistics
McGill University
805 Sherbrooke West
Montreal, QC H3A 2K6
Canada