A geometric approach to the diophantine Frobenius problem

TL;DR

This paper reveals a geometric structure underlying the Frobenius problem for three coprime integers, enabling efficient computation of key quantities like the largest non-representable number and its count.

Contribution

It introduces a geometric framework that simplifies and accelerates the calculation of Frobenius problem solutions for three coprime integers.

Findings

Derived a formula for the largest non-representable number.

Provided a method to compute the count of non-representable numbers.

Achieved an O(log(max a_i)) computational complexity.

Abstract

It turns out that all instances of the diophantine Frobenius problem for three coprime a_i have a common geometric structure which is independent of arithmetic coincidences among the a_i. By exploiting this structure we easily obtain Johnson's formula for the largest non-representable z, as well as a formula for the number of such z. A procedure is described which computes these quantities in O(log(max a_i)) steps.