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

Nonnegative errors

The scalar update already settles nonnegative errors. With indices starting at zero, $C_{n + 1} = C_{n} - E_{n}$ makes $C_{n}$ nonincreasing, and a decreasing sequence of natural numbers must stabilise.

Theorem 8.1 (descent). Let $C, E : N \to N$ satisfy $C_{n + 1} + E_{n} = C_{n}$ for every $n$ . Then $E_{n} = 0$ for all sufficiently large $n$ .

Proof. The recurrence gives $C_{n + 1} \leq C_{n}$ . Choose an index at which the natural-number sequence reaches its minimum. Every later value equals that minimum, and $C_{n + 1} + E_{n} = C_{n}$ then gives $E_{n} = 0$ . ◻

The hypothesis is exactly one-sidedness: $C$ and $E$ take values in $N$ . If $E_{n} < 0$ infinitely often, the numerator rises at those indices and this descent argument no longer applies. The next two sections first treat constant and periodic negative errors.

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.

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