On a problem of Frobenius in three numbers

TL;DR

This paper introduces an efficient algorithm for computing the Frobenius number for three pairwise coprime positive integers, leveraging the Chinese remainder theorem to achieve constant average runtime.

Contribution

It presents a novel algorithm with constant average time complexity for finding the Frobenius number in three integers, improving computational efficiency.

Findings

Algorithm has O(1) average running time.

Enables direct computation of Frobenius number for three integers.

Utilizes Chinese remainder theorem for efficient calculation.

Abstract

For three positive integers ai, aj, ak pairwise coprime, we present an algorithm that find the least multiple of ai that is a positive linear combination of aj, ak. The average running time of this algorithm is O(1). Using this algorithm and the chinese remainder theorem leads to a direct computation of the Frobenius number f(a1, a2, a3).