Reciprocal Sums and the Sylvester Recurrence

Will Cook

Short paper

Reciprocal Sums and the Sylvester Recurrence

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

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Summary. For strictly increasing positive integers with $a_{n}^{2} / a_{n + 1} = 1 + 3 / n + o (n^{- 3})$ , the reciprocal sum is irrational. The argument extracts a cubic polynomial from an integer tail numerator, restricts its coefficients using a square in a cubic field, and excludes the remaining cases modulo seven. Under rationality and the quadratic growth limit, separate criteria using bounded numerator increases and weighted first crossings force the eventual Sylvester recurrence. The unrestricted Erdős–Graham question remains open.

This paper covers cubic-rate irrationality, nonintegral regular-rate exclusion and sufficient tail criteria for the Sylvester recurrence; the unrestricted Erdős–Graham question remains open.

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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Sources and related work

The companion Reciprocal Sums and the Sylvester Recurrence: Further Results and Proofs supplies full proofs and fixed-revision links: Section 2 for regular rates and cubic disagreement; Sections 4–5 for integer tails; Section 12 for the gcd and bounded-increment arguments; Section 6 for LCM results; Section 13 for scalar summability; and Section 14 for open estimates and failed constructions. A final map records dependencies. The auxiliary constructions satisfy only their stated hypotheses, not necessarily those of a rational reciprocal series.

For the earlier integer-remainder method see Erdős–Straus [12]. The companion’s Section 3 distinguishes polynomial Cantor series [13] and the freely selectable series of Problem #270 [15] from the present setting. The Isabelle/HOL work [16] formalises older Erdős–Straus criteria, not Problem #243. General-rate extraction uses the Gamma-ratio asymptotics in [17]; the cubic case uses the explicit $F_{n}$ .

Indexing of the canonical tails.

Section 2 starts with $C_{1} = p$ , $D_{1} = q$ . Setting $a_{0} = 1$ , $D_{0} = q$ , $C_{0} = p + q$ extends the recurrences by one update, but gives neither $a_{0} > 1$ nor strict increase at that step. The zero-based formal cubic theorem instead uses $N$ with $a_{N} > N$ , $b_{n} = n + 1$ for $0 \leq n < N$ and $b_{n} = a_{n}$ thereafter. This sequence is strictly increasing and changes the reciprocal sum by a rational finite sum without shifting $n$ or $3 / n$ on the tail. See the companion’s indexing subsection.

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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