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

The direct analytic question

Problem 14.5 (summability from rationality). Let $(a_{n})$ satisfy the hypotheses of Problem 1.1, and let $(D_{n}, C_{n}, E_{n})$ be its integer tail. Must

\sum_{n = 0}^{\infty} \frac{(- E_{n})_{+}}{C_{n}} < \infty ?

An affirmative answer closes Problem #243 by Theorem 13.1. Since the summands tend to zero and $\log (1 + t) \sim t$ , convergence is equivalent to bounded partial products $\prod_{n < N} (1 + (- E_{n})_{+} / C_{n})$ . A negative answer requires the full growth and rationality hypotheses, not merely a locally admissible orbit. By Theorem 6.2, finiteness of (21) for one nonincreasing weight with divergent integral and one fixed $B \geq 0$ also suffices. The equivalent record-only sum in Theorem 6.2 weights the positive excess of each actual upward jump over $B$ by $f (U_{n})$ .

Remark. One further question is not left open here, since it is answered in [12]: whether vanishing relative error follows from $a_{n + 1} / a_{n}^{2} \to 1$ and rationality. It does. After deleting a finite prefix, Koizumi’s Corollary 3 [12] makes the sequence the pseudo-greedy expansion of its own reciprocal sum and gives $ε_{n} \to 0$ under exactly the hypotheses of Problem 1.1; rationality is not needed for that step, only summability of $\sum 1 / a_{n}$ . The matched-index dictionary in Section 3 gives $ε_{n} = E_{n} / C_{n}$ on the restarted tail. For a rational sum, Lemma 4(2) [12] also gives $- c_{n} / 2 \leq e_{n} < c_{n} / 2$ at every index of that expansion. Thus $| E_{n} | < C_{n}$ holds beyond the restart, as required by Theorem 12.1; it need not hold at every original index. The eventual inequality $| E_{n} | < C_{n}$ also follows directly from vanishing relative error with $K = 1$ . What remains unsupplied is the eventual lower bound on $E_{n}$ in Theorem 12.1. That additional hypothesis is why the theorem is still conditional.

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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