Automated Conjecturing of Frobenius Numbers via Grammatical Evolution

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 faster conjecturing but also from finding conjectures previously out of our grasp. Grammatical evolution (GE), 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 GE to it. In this manner, a few formulas for Frobenius numbers of specific quadruples were found automatically. The sketch of the proof of one conjectured formula, using lattice point enumeration method, is provided as well. The same method can easily be used on other problems to speed up and enhance the research process.

KEYWORDS:

• automated conjecturing
• Frobenius problem
• grammatical evolution

2010 AMS Subject Classification:

• 11D07
• 68Q42