Automatic Solving of Cubic Diophantine Equations Inspired by Ramanujan

TL;DR

This paper develops an algorithm inspired by Ramanujan's identities to automatically solve a broad class of cubic Diophantine equations, combining historical insights and modern automation.

Contribution

It introduces a novel algorithm that automates and generalizes Ramanujan-inspired methods for solving cubic Diophantine equations.

Findings

Successfully solves a large class of cubic Diophantine equations.

Automates the discovery of polynomial identities related to Ramanujan's work.

Extends previous theoretical approaches through computational methods.

Abstract

In Ramanujan's Lost Notebook there is an amazing identity that furnishes infinitely many "almost counterexamples" to the cubic Fermat's Last Theorem, with no indication whatsoever how he discovered it. In 1995, Michael Hirschhorn explained, in a brilliant way, how Ramanujan may have done it, based on a certain polynomial identity for a sum of four cubes. Much earlier, Eri Jabotinsky, in an article published in 1946 (in a mathematics journal for teenagers) explained how Ramanujan may have discovered these polynomial identities needed for Hirschhorn's approach. Here we combine these two brilliant ideas (that may or may not have been how Ramanujan did it), automate it, and generalize, by developing an algorithm to solve a large class of cubic diophantine equations. Our interest in this problem was rekindled after reading Amy Alznauer's (b. Andrews) delightful children book "The Boy Who Dreamed of Infinity" (Candlewick Press), 2020, where Ramanujan's identity appears in one of the illustrations.