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

Introduction

Sylvester’s sequence $2, 3, 7, 43, 1807, \dots$ has reciprocal sum $1$ . Its recurrence $a_{n} = a_{n - 1}^{2} - a_{n - 1} + 1$ gives the telescoping identity

\frac{1}{a_{n - 1} - 1} = \frac{1}{a_{n - 1}} + \frac{1}{a_{n} - 1} .

The tail beginning with $1 / a_{n - 1}$ therefore equals $1 / (a_{n - 1} - 1)$ . Erdős Problem #243 asks whether every rational reciprocal sum with $a_{n} \sim a_{n - 1}^{2}$ must eventually have this form:

Problem 1.1 (Erdős #243). Let $1 \leq a_{1} < a_{2} < \dots$ be a sequence of integers with

lim_{n \to \infty} \frac{a_{n}}{a_{n - 1}^{2}} = 1 and \sum \frac{1}{a_{n}} \in Q .

Then $a_{n} = a_{n - 1}^{2} - a_{n - 1} + 1$ for all sufficiently large $n$ .

See [5] and [6]. Bloom’s catalogue, accessed 28 July 2026, lists the problem as open [13]. Below we index the same recurrence as $a_{n + 1} = a_{n}^{2} - a_{n} + 1$ , which is the shift the formal sources use; the two forms are the same statement. The polynomial $a^{2} - a + 1$ is called the Sylvester successor.

Clearing a rational reciprocal tail gives positive integers $C_{n}, D_{n}$ with $C_{n + 1} = a_{n} C_{n} - D_{n}$ and $D_{n + 1} = a_{n} D_{n}$ . The first update can also be written $C_{n + 1} = C_{n} - E_{n}$ , where $E_{n} = D_{n} - (a_{n} - 1) C_{n}$ . Quadratic growth makes $E_{n} / C_{n} \to 0$ . The eventual Sylvester recurrence follows once the integer error itself vanishes.

There are two ways to use this integer numerator. At a precise regular rate it grows, but a high finite difference tends to zero and must vanish. The resulting polynomial is then tested against the denominator recurrence. With bounded upward steps, a different argument applies: a stable common divisor gives reduced tails, whose numerators avoid every earlier multiplier. The Chinese remainder theorem produces a block that bounded rises cannot cross, regardless of intervening decreases. Neither argument replaces the denominator recurrence by a scalar growth estimate.

Section 2 proves cubic and nonintegral-rate irrationality: these rates exclude the eventual Sylvester recurrence. The general question remains open. After the literature comparison in Section 3, Sections 4–5 develop integer tails and zero-error identities for the subsequent criteria.

The later criteria use two distinct normalisations. Section 6 clears tails with an LCM: common numerator factors remain, so congruences forbid short crossings rather than numerator values. Section 7 reduces to lowest terms, where cancellation can remove a prime that persists in the LCM. That argument must track the prime powers which survive. Sections 9–10 treat constant and periodic errors; Sections 11–12 prove the bounded-increase criterion, and Section 13 treats scalar summability. Section 14 separates open estimates from countermodels to weakened hypotheses. The appendices give formal sources, the finite residue calculation and a result map.

We use $z_{+} = max (z, 0)$ . Deleting a finite prefix is harmless for an eventual recurrence, but not for the coefficients in a higher-order rate. In particular, the index $n$ in $1 + 3 / n + o (n^{- 3})$ is kept fixed throughout the cubic proof. The recurrence sections also use zero-based indexing, with that convention stated where the integer sequences are introduced.

Keywords. irrationality; Ahmes series; Sylvester’s sequence; unit fractions; Lean 4. MSC 2020. 11J72 (primary); 11B37, 11D68, 68V20 (secondary).

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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