On the Frobenius number for three variables

TL;DR

This paper addresses and corrects minor inconsistencies in an existing algorithmic formula for computing the Frobenius number of three positive integers with no common divisor.

Contribution

The paper provides a corrected and consistent version of Tripathi's 2017 algorithmic formula for the Frobenius number in three variables.

Findings

Resolved inconsistencies in the previous formula

Provided a verified, consistent formula for three-variable Frobenius numbers

Enhanced accuracy of Frobenius number computations for three integers

Abstract

For positive integers $a$, $b$, and $c$ which have no common divisor, the Frobenius number of $a$, $b$ and $c$ is defined to be the largest integer that cannot be expressed as a linear combination of $a$, $b$ and $c$ with non-negative integer coefficients. In 2017, Tripathi gave an algorithmic formula for the Frobenius number in three variables, however there were some minor inconsistencies in the formula. In this paper, we settle these inconsistencies.