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

  • Andres Ventas
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Abstract

We introduce the Ceiling Continued Fractions (FCT) framework for constructing three-term Egyptian fraction representations in the Erdős--Straus conjecture from divisors of the shifted integers p+i. A probabilistic model for their arithmetic structure leads to an asymptotic upper-bound heuristic for cumulative failure. The resulting bound is summable, and the first Borel--Cantelli lemma assigns probability zero to the event that infinitely many primes survive the full FCT block. Computational tests on 109 Mordell-type primes at each of the scales 1017, 1052, and 10131 found no counterexamples.

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Cite

BibTeXRIS
BibTeX
@misc{ventas2026ceiling,
  title = {{A Ceiling Continued Fraction Approach to the Erdős-Straus Conjecture: Heuristic finiteness of counterexamples}},
  author = {Andres Ventas},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.04551},
  eprint = {2605.04551},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.04551},
}
RIS
TY  - UNPB
TI  - A Ceiling Continued Fraction Approach to the Erdős-Straus Conjecture: Heuristic finiteness of counterexamples
AU  - Andres Ventas
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2605.04551
UR  - https://arxiv.org/abs/2605.04551
AB  - We introduce the Ceiling Continued Fractions (FCT) framework for constructing three-term Egyptian fraction representations in the Erdős--Straus conjecture from divisors of the shifted integers p+i. A probabilistic model for their arithmetic structure leads to an asymptotic upper-bound heuristic for cumulative failure. The resulting bound is summable, and the first Borel--Cantelli lemma assigns probability zero to the event that infinitely many primes survive the full FCT block. Computational tests on 10^9 Mordell-type primes at each of the scales 10^17, 10^52, and 10^131 found no counterexamples.
ER  -