A Ceiling Continued Fraction Approach to the Erdős-Straus Conjecture: Heuristic finiteness of counterexamples

Abstract

We introduce the Ceiling Continued Fractions (FCT) framework for constructing three-term Egyptian fraction representations in the Erdős-Straus conjecture. The approach exploits divisor structures of shifted integers p+i rather than congruence-based techniques. 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. Computational tests on 10^9 primes in ranges around 10^17, 10^52, and 10^131, show no counterexamples with very small search depth.