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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Further questions

We collect the restrictions on a possible counterexample and the estimates which would exclude it. The constant and periodic arguments concern all-negative tails. They leave open a periodic pattern seen only along the negative indices of a mixed-sign sequence. The unrestricted problem remains unresolved here.

Proposition 14.1 (necessary conditions on a counterexample). For the integer tail attached to any counterexample to Problem 1.1,

E_{n} \neq 0 eventually, \frac{| E_{n} |}{C_{n}} ⟶ 0,

and

\underset{\begin{matrix} n \to \infty \\ E_{n} < 0 \end{matrix}}{lim sup} (- E_{n}) = \infty, \sum_{n = 0}^{\infty} \frac{(- E_{n})_{+}}{C_{n}} = \infty .

In particular, negative indices occur infinitely often. Thus the remaining regime consists of unbounded negative errors along exact reciprocal tails for which the sum of relative increases diverges.

Proof. Koizumi’s construction gives vanishing relative error after a finite shift, hence $| E_{n} | < C_{n}$ eventually. Absorption then makes $E_{n} \neq 0$ eventually, since eventual zero would give the Sylvester recurrence. Descent excludes an eventually nonnegative error. Theorem 12.1 excludes a bounded negative part, and Theorem 13.1 excludes convergence of the sum of relative increases. These statements are unchanged by deleting a finite prefix. ◻

Proposition 14.1 does not assert infinitely many prime divisors of the error magnitudes. Nor are its numerical conditions alone a reformulation of the problem: the numerator and denominator must satisfy the exact recurrences. With those recurrences, positivity and vanishing relative error, Section 4 supplies the reciprocal-series realisation. The following exact-orbit question is therefore equivalent to the original problem.

Problem 14.2 (unbounded negative errors with divergent relative sum). Exclude, or construct, natural-number sequences $(a, D, C)$ satisfying (19), with $a_{n} > 1$ and $C_{n} > 0$ , whose multipliers satisfy $lim_{n} a_{n + 1} / a_{n}^{2} = 1$ and whose error satisfies vanishing relative error, and which is negative infinitely often with magnitudes unbounded along that infinite set and

\sum_{n} \frac{(- E_{n})_{+}}{C_{n}} = \infty .

Such an orbit fails the bounded-negative-part and summable-relative-increase hypotheses. By Proposition 14.1 the integer tail of a counterexample is such an orbit after deletion of a finite prefix, so an exclusion would settle Problem #243. Conversely, a global orbit with the displayed properties has $C_{n} / D_{n} \to 0$ by Section 4. The reciprocal-series realisation then identifies its reciprocal sum as $C_{0} / D_{0}$ . After deletion of a finite prefix the multipliers are strictly increasing, and the infinitely many negative errors rule out a Sylvester tail. It is therefore a counterexample to Problem #243. Finite admissible prefixes alone do not provide such a construction.

Remark. The recurrences alone admit unbounded negative errors for which $\sum_{n} (- E_{n})_{+} / C_{n}$ diverges. The following example fails both the relative-error and growth assumptions of Problem 14.2. Put

C_{n} = 2^{n}, b_{0} = 2, b_{n + 1} = \frac{1}{2} b_{n} (b_{n} + 2), a_{n} = b_{n} + 2, D_{n} = b_{n} C_{n},

so that $(b_{n}) = 2, 4, 12, 84, 3612, \dots$ and $(a_{n}) = 4, 6, 14, 86, 3614, \dots$ . Every $b_{n}$ is a positive even integer, since $b_{n} = 2 k$ gives $b_{n + 1} = 2 k (k + 1)$ , so all four sequences are $N$ -valued. They satisfy both recurrences: $C_{n + 1} + D_{n} = 2^{n} (2 + b_{n}) = a_{n} C_{n}$ and $D_{n + 1} = b_{n + 1} 2^{n + 1} = b_{n} (b_{n} + 2) 2^{n} = a_{n} D_{n}$ . Its error is

E_{n} = D_{n} - (a_{n} - 1) C_{n} = b_{n} 2^{n} - (b_{n} + 1) 2^{n} = - C_{n},

so the negative magnitudes $2^{n}$ are unbounded and $\sum_{n} (- E_{n})_{+} / C_{n}$ diverges. Here $| E_{n} | = C_{n}$ at every index, so both the strict inequality $| E_{n} | < C_{n}$ and the vanishing relative error fail. The multipliers satisfy $a_{n + 1} = \frac{1}{2} (a_{n}^{2} - 2 a_{n} + 4)$ , so $a_{n + 1} / a_{n}^{2} \to \frac{1}{2}$ and the growth hypothesis fails as well. The example therefore says nothing about Problem 1.1, and it does not isolate the error bounds: $E_{n} = - C_{n}$ at every index forces that multiplier recurrence, so growth fails with them. It is recorded only to show that the two recurrences alone do not force the required conclusion.

A scalar profile need not satisfy the recurrences.

Growth of the factors repeated in the denominator

Cancellation during recovery intervals

The direct analytic question

Unbounded upward increments

Formalisation targets

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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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