Reciprocal Sums and the Sylvester Recurrence: Further Results and Proofs

Will Cook

Long paper

Reciprocal Sums and the Sylvester Recurrence: Further Results and Proofs

Erdős #243 By Will Cook September 2026 Authorship, AI use and citation

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Summary. Integer finite differences exclude nonintegral regular rates and reduce the cubic rate to a polynomial. A square condition in the cubic field and congruences modulo seven then prove irrationality. For rational reciprocal sums with $a_{n + 1} / a_{n}^{2}$ tending to one, the record derives sufficient Sylvester-recurrence criteria from persistent divisors, bounded numerator increases and weighted crossings of new maxima. The arithmetic estimates needed to apply these criteria to every rational reciprocal sum remain open.

This paper covers cubic and nonintegral regular-rate irrationality, integer-tail arithmetic and sufficient Sylvester-recurrence criteria, with their separate hypotheses and remaining estimates.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #243, which remains open.

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In this paper

Result map and proof dependencies

Beyond the short paper, this companion supplies signed-series comparisons (Section 3), integer coefficients (Section 6), general $1 + c / n$ rates (Section 7), and regular-rate extraction with positive-density cubic exclusion (Section 2). Theorem 7.3 gives the double-logarithmic bound; Section 14 includes the scalar failed-divisibility example. The principal dependencies are:

Bounded upward increments.

Vanishing relative error gives eventual absorption. Bounded negative values at arbitrarily late indices give a stable gcd; CRT then excludes the unbounded reduced tail with bounded upward increments. The increment bound is an extra hypothesis.

A convergent sum of relative increases.

The inequality $C_{n + 1} \leq C_{n} (1 + (- E_{n})_{+} / C_{n})$ turns summability into a uniform bound on $C_{n}$ . Integrality then permits only finitely many rises, after which the numerator stabilises. No vanishing relative error is needed.

New maxima and prime-power persistence.

The amplified error bounds cancellation, protecting large primes for Theorem 7.5. Density-one coprimality and CRT height control give the double-logarithmic bound. LCM tails give weighted crossings without primitive numerators. For reduced tails, Corollary 7.2 protects a prime power below its numerator threshold, allowing crossings to be counted before that protection fails.

Cubic rate.

The tail estimate and finite differences of the Gamma ratio force an eventual cubic polynomial for the integer numerator. The recurrence forces a modular square condition, which the square-specialisation lemma lifts to the cubic field. The ordinary proof uses Chebotarev; the Lean proof obtains the same implication from a Dedekind-zeta pole comparison. The trace calculation leaves $m = 12$ , and both possible signs are excluded modulo seven.

Avoidance of fixed moduli.

For sufficiently sparse prime moduli, a coprime integer sequence can have unbounded but very small increments. At the specified double-exponential scale, counting residue classes and applying CRT give the largest gaps between integers divisible by none of the whole moduli. These auxiliary results do not construct reciprocal-tail examples.

What remains.

For the unrestricted problem, one still needs the stated summability or record bound, or a contradiction to the necessary conditions on a counterexample. The static examples and local identities supply neither.

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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