A Fast Algorithm for Denumerants with Three Variables

TL;DR

This paper introduces a highly efficient algorithm with logarithmic time complexity for computing the denumerant function in three variables, significantly improving computational speed for this problem.

Contribution

The paper presents a novel algorithm that computes the denumerant for three variables in logarithmic time, enhancing previous methods' efficiency.

Findings

Algorithm computes denumerants in O(log b) time

Significant speed-up over existing algorithms

Applicable to large integers with high efficiency

Abstract

Let $a,b,c$ be distinct positive integers such that $a<b<c$ and $\gcd(a,b,c)=1$. For any non-negative integer $n$, the denumerant function $d(n;a,b,c)$ denotes the number of solutions of the equation $ax_1+bx_2+cx_3=n$ in non-negative integers $x_1,x_2,x_3$. We present an algorithm that computes $d(n;a,b,c)$ with a time complexity of $O(\log b)$.