On the Frobenius Coin Problem in Three Variables

TL;DR

This paper introduces a new, faster algorithm for solving the Frobenius coin problem in three variables, significantly improving efficiency over existing methods with a logarithmic time complexity.

Contribution

The paper presents a novel algorithm that solves the Frobenius coin problem in three variables more efficiently than previous algorithms, with a complexity of O(log a_1).

Findings

Algorithm runs in O(log a_1) steps.

Faster and simpler than existing algorithms.

Applicable to three-variable Frobenius coin problem.

Abstract

The Frobenius coin problem in three variables, for three positive relatively prime integers $a_1< a_2< a_3$ asks to find the largest number not representable as $a_1x_1+a_2x_2+a_3x_3$ with non-negative integer coefficients $x_1$, $x_2$ and $x_3$. In this article, we present a new algorithm to solve this problem that is faster and in our belief simpler than all existing algorithms and runs in $\mbox{O}(\log a_1)$ steps.