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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In this paper

Introduction

The Sylvester recurrence $a_{n + 1} = a_{n}^{2} - a_{n} + 1$ gives a rational reciprocal sum by telescoping:

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

For an increasing sequence of positive integers that eventually satisfies this recurrence, the tail sum at every sufficiently large index is $1 / (a_{n} - 1)$ . Erdős and Graham asked whether every rational reciprocal sum with $a_{n + 1} \sim a_{n}^{2}$ arises in this way.

Problem 1.1 (Erdős Problem 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 $n$ .

Erdős and Graham [5] asked this question; see also Erdős [6] and Bloom’s catalogue [11]. We prove the following irrationality result for a restricted class of sequences with $a_{n + 1} \sim a_{n}^{2}$ .

Theorem 1.2 (cubic-rate irrationality). A strictly increasing sequence of positive integers with

a_{n}^{2} / a_{n + 1} = 1 + \frac{3}{n} + o (n^{- 3})

has irrational reciprocal sum.

The Sylvester recurrence gives $a_{n}^{2} / a_{n + 1} - 1 = O (1 / a_{n})$ , so the cubic-rate hypothesis excludes that behaviour. It imposes much more than the quadratic limit in Problem 1.1.

For example, set $a_{1} = 8$ and

a_{n + 1} = ⌈ \frac{n a_{n}^{2}}{n + 3} ⌉ .

This gives $8, 16, 103, 5305, \dots$ . From $a_{n + 1} \geq a_{n}^{2} / 4$ we obtain $a_{n} \geq 4 \cdot 2^{2^{n - 1}}$ by induction. The error introduced by the ceiling is less than one, and hence

0 \leq 1 + \frac{3}{n} - \frac{a_{n}^{2}}{a_{n + 1}} < \frac{16}{a_{n}^{2}} = o (n^{- 3}) .

The theorem applies to this explicitly defined sequence.

Koizumi’s product criterion [10] requires the increments of $P_{n} / a_{n}$ , where $P_{n} = \prod_{j < n} a_{j}$ , to have nonpositive upper limit. Corollary 4.9 below allows any finite upper limit. At the cubic rate, however, $P_{n} / a_{n}$ grows like $n^{3}$ and its increments like $n^{2}$ ; neither criterion applies. This comparison does not decide whether the LCM criteria apply: those also depend on repeated factors in $P_{n}$ .

Rationality supplies a positive integer tail numerator $C_{n}$ , but not a bounded one: here it grows cubically. We therefore apply integrality to its finite differences. The exact comparison sequence $F_{n} = n (n + 1) (n + 2)$ has $F_{n + 1} / F_{n} = 1 + 3 / n$ . The precision in the hypothesis forces $Δ^{4} C_{n} \to 0$ , hence eventual equality $C_{n} = A F_{n} + B$ . This extraction uses the numerator ratio and integrality, not the exact denominator recurrence.

The denominator recurrence now becomes decisive. It imposes a square condition at every root of a cubic modulo almost every prime. Chebotarev’s theorem turns these residue conditions into a square in a cubic number field, where traces restrict the numerator to two polynomials. A four-term calculation modulo seven excludes both. Section 3 follows these two stages: first extract the polynomial, then test whether it can satisfy the exact recurrence.

Sections 4 and 5 give bounded-increment and summability criteria. Section 6 clears tails by a least common multiple and uses congruences to control first crossings of new heights; its convergence hypothesis is not derived from the quadratic limit. The companion paper also treats nonintegral rates and proves a stronger cubic disagreement theorem. Its proof locations and the formal sources are listed in Appendix B.

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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