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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Finite total relative increase

We can bound a positive integer sequence by multiplying the relative increases of its terms. If their sum converges, that product is finite. Only finitely many upward integer steps can then occur, and descent finishes the argument. No denominator recurrence is needed. For comparison, $C_{n} = n + 1$ and $C_{n} = (n + 1)^{2}$ have relative increases tending to zero but divergent sums. The eventual Sylvester recurrence gives an eventually constant numerator, so it satisfies the summability condition.

Theorem 13.1 (finite sum of relative increases). Let $C_{n} \in N_{> 0}$ and $E_{n} \in Z$ satisfy $C_{n + 1} = C_{n} - E_{n}$ . If

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

then $E_{n} = 0$ eventually. For an exact reciprocal-tail orbit this implies $a_{n + 1} = a_{n}^{2} - a_{n} + 1$ eventually. No hypothesis $| E_{n} | / C_{n} \to 0$ is required.

Proof. Put $δ_{n} = (- E_{n})_{+} / C_{n}$ . The update gives $C_{n + 1} \leq C_{n} (1 + δ_{n})$ , so

C_{N} \leq C_{0} \prod_{n < N} (1 + δ_{n}) \leq C_{0} \exp (\sum_{n} δ_{n}) .

Choose an integer upper bound $K$ for $C_{n}$ . Each strict rise of the integer sequence contributes at least $1 / K$ to $\sum_{n} δ_{n}$ , so there are only finitely many rises. The remaining positive integer sequence is nonincreasing and therefore stabilises, forcing $E_{n} = 0$ . The recurrence follows from Theorem 5.7, since $C_{n} > 0$ . ◻

Integer increments suffice even for positive real $C_{0}$ : the bounded values lie in the finite set $(C_{0} + Z) \cap (0, K]$ , so the same proof works. The Lean statement retains integer-valued $C_{n}$ .

The two indispensable features of this argument are positivity and discrete increments. With $C_{n} = 1 + 1 / (n + 1)$ and $E_{n} = 1 / ((n + 1) (n + 2))$ , the numerator decreases forever with zero relative-increase sum, but the errors are not integers. With $C_{n} = - n - 1$ and $E_{n} = 1$ , the increments are integers but positivity fails.

The pseudo-greedy gap version, with the same product bound and integer descent, appears in the 12 August 2026 working report [9]. Bado additionally accounts for factors removed in LCM clearing [8]. In the notation of Section 6, the update gives

\log U_{N} \leq \log U_{0} + \sum_{n < N} \frac{(- E_{n})_{+}}{C_{n}} - \sum_{n < N} \log ρ_{n} .

His sufficient condition is an upper bound on the difference of these two sums. The subtractive term records the decrease caused by a repeated factor in the denominator. It may offset some relative increases; the hypothesis does not require either sum to converge separately. Unlike the scalar criterion above, this argument also uses the pseudo-greedy limit: bounded $U_{n}$ and $V_{n} / U_{n} = E_{n} / C_{n} \to 0$ make the integer $V_{n}$ eventually zero, after which $ρ_{n} U_{n + 1} = U_{n}$ gives eventual constancy.

Corollary 3 [12] and Lemma 4 [12] supply the same positive integer update after a finite restart. Theorem 13.1 therefore applies if the relative-increase sum converges. This is the sole additional hypothesis, asked for in Problem 14.5; the cited construction does not establish it.

Example 13.2 (the product bound at a geometric rate). Suppose $C_{0} = 100$ and $(- E_{n})_{+} / C_{n} \leq 2^{- n - 1}$ for every $n$ , so that the sum of relative increases is at most $1$ ; the constraint at $n = 0$ still permits $E_{0}$ as negative as $- 50$ . Since $1 + x \leq e^{x}$ ,

C_{N} \leq 100 \prod_{n < N} (1 + 2^{- n - 1}) \leq 100 e < 272

for every $N$ . For all sufficiently large $n$ , we have $2^{- n - 1} < 1 / 272$ . A strict rise would then force $(- E_{n})_{+} / C_{n} \geq 1 / C_{n} > 1 / 272$ , a contradiction. Thus $C_{n}$ is eventually nonincreasing; as a positive integer sequence it stabilises, and $E_{n} = 0$ thereafter. The constant $272$ comes from the total mass and not from any single magnitude, which is why the hypothesis of Theorem 13.1 is summability of $(- E_{n})_{+} / C_{n}$ and not a bound on it.

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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