Some Experimental Results on the Frobenius Problem

Abstract

We study the Frobenius problem: Given relatively prime positive integers {\small $a_{1} , \dots , a_{d}$}, find the largest value of {\small $t$} (the Frobenius number) such that {\small $ \sum_{k=1}^d m_{k} a_{k} = t $} has no solution in nonnegative integers {\small $ m_{ 1 } , \dots , m_{ d } $}. Based on empirical data, we conjecture that except for some special cases, the Frobenius number can be bounded from above by {\small $ \sqrt{a_{1}a_{2}a_{3}}^{5/4} - a_1 - a_2 - a_3 $}.