Automated conjecturing of Frobenius numbers via grammatical evolution

TL;DR

This paper demonstrates how grammatical evolution, a genetic programming technique, can automate the conjecturing of formulas for Frobenius numbers, potentially accelerating discoveries in number theory.

Contribution

It introduces interpreting the Frobenius problem as a symbolic regression task and applies grammatical evolution to automatically find formulas for specific cases.

Findings

Automatically found formulas for Frobenius numbers of certain quadruples

Provided a proof sketch for one conjectured formula using lattice point enumeration

Method can be generalized to other mathematical problems

Abstract

Conjecturing formulas and other symbolic relations occurs frequently in number theory and combinatorics. If we could automate conjecturing, we could benefit not only from speeding up, but also from finding conjectures previously out of our grasp. Grammatical evolution, a genetic programming technique, can be used for automated conjecturing in mathematics. Concretely, this work describes how one can interpret the Frobenius problem as a symbolic regression problem, and then apply grammatical evolution to it. In this manner, a few formulas for Frobenius numbers of specific quadruples were found automatically. The sketch of the proof for one conjectured formula, using lattice point enumeration method, is provided as well. Same method can easily be used on other problems to speed up and enhance the research process.