A Ceiling Continued Fraction Approach to the Erd\H{o}s-Straus Conjecture: Heuristic finiteness of counterexamples

TL;DR

This paper introduces a novel ceiling continued fraction method for the Erdős–Straus conjecture, providing heuristic evidence that counterexamples are likely finite and demonstrating no small counterexamples in extensive computational tests.

Contribution

The paper develops a new ceiling continued fraction framework that exploits divisor structures, offering heuristic finiteness evidence for potential counterexamples.

Findings

No counterexamples found up to very large bounds in extensive computational tests.

Derived a super-polynomial upper bound on the failure probability.

Heuristic arguments suggest counterexamples, if any, are finite.

Abstract

We introduce the Ceiling Continued Fractions (FCT) framework for constructing three-term Egyptian fraction representations in the Erdos-Straus conjecture. The approach exploits divisor structures of shifted integers p+i rather than congruence-based techniques. Computational tests on 10^9 primes in ranges around 10^17 and 10^52, and 10^7 primes around 10^131, show no counterexamples with very small search depth. We derive a super-polynomial upper bound on the failure probability; its convergence, together with the Borel-Cantelli lemma, provides heuristic evidence that counterexamples, if any exist, form a finite set.