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

Counting jumps before a prime power can be lost

Large jumps may skip forbidden values. Under the relative-error bound, $L = p Q / 2$ keeps the numerator before cancellation below $p Q$ until the crossing, protecting the prime power. Odd multiples of $p$ exclude two-unit crossings as well: skipping an odd level in two units would leave both adjacent numerators even. We count both crossing jumps and their excess over two.

Theorem 7.9 (counting crossings before a prime power is lost). Let the orbit satisfy the reduced recurrences of this section, with $2 | {\tilde{e}}_{n} | < u_{n}$ from an index $s$ . Let $p \geq 3$ be prime, $Q = p^{ℓ}$ divide $v_{s}$ with $Q \geq 16$ , put $L = p Q / 2$ , and assume $R_{s} < L / 2$ . Assume that $u_{t} \geq L$ for some $t > s$ , and let $τ$ be the first such index, let $J$ be the set of steps in $[s, τ)$ that first cross at least one odd multiple of $p$ in $(L / 2, L]$ , and put $X = \sum_{n \in J} (d_{n} - 2)$ . Then every $n \in J$ is a record step with $h_{n} = 1$ and $d_{n} \geq 3$ , and $p Q \leq (8 p + 8) | J | + 4 X + 8 p$ .

Proof of Theorem 7.9. For $s \leq n < τ$ , one has $u_{n} < L$ and $w_{n} < 3 u_{n} / 2 < 3 p Q / 4 < p^{ℓ + 1}$ , since $Q = p^{ℓ}$ . Corollary 7.2 gives $Q ∣ v_{t}$ throughout $[s, τ]$ , so $p ∤ u_{t}$ there. A first crossing of a level above $R_{s}$ is a record step. If $h_{n} \geq 2$ then $u_{n + 1} = w_{n} / h_{n} < 3 u_{n} / 4$ , so every such record step has $h_{n} = 1$ . An odd multiple of $p$ cannot be landed on. A jump of size $1$ crossing it would land on it; a jump of size $2$ avoiding the landing would have two even endpoints, contrary to $gcd (u_{n}, u_{n + 1}) = 1$ . Thus every $n \in J$ has $d_{n} \geq 3$ . It remains to count these levels. The interval $(L / 2, L]$ has length $p Q / 4$ , and odd multiples of $p$ are spaced by $2 p$ , so it contains at least $Q / 8 - 1$ of them. The levels first crossed by a step of size $d_{n}$ have the same spacing, so there are at most $1 + d_{n} / (2 p)$ of them. Summing over $J$ gives

Q / 8 - 1 \leq | J | + \frac{2 | J | + X}{2 p},

which rearranges to the asserted inequality. ◻

Crossing requires many jumps or a few large ones. The first series therefore counts jumps of size at least three and measures their excess over two, with weights that detect either contribution. A non-Sylvester tail pays a fixed positive amount on arbitrarily late protected intervals; a Sylvester tail has no late records. The relative-error limit alone gives no convergence.

Theorem 7.10 (two convergence criteria for large record jumps). Under the standing hypotheses, the sum $E = \sum_{n record} (1_{d_{n} \geq 3} u_{n}^{- 1 / 2} + (d_{n} - 2)_{+} u_{n}^{- 1})$ is finite if and only if the sequence is eventually Sylvester; and $\sum_{n record} (d_{n} - 2)_{+} u_{n}^{- 1 / 2}$ is finite if and only if the sequence is eventually Sylvester.

Proof. Assume first that the recurrence is not eventually Sylvester. Then $u_{n} \to \infty$ . For every sufficiently late $s$ , Lemma 7.7 with $A = 3$ gives an odd $Q = p^{k} ∣ v_{s}$ with $Q > (H_{s} + 2)^{3}$ , hence $Q \geq 16$ and $p Q > 4 R_{s}$ . Set $L = p Q / 2$ and take its first crossing. The preceding theorem applies. Because $Q \geq p$ and $p \geq 3$ ,

\frac{8 p}{L} = \frac{16}{Q} \leq 1, \frac{8 p + 8}{\sqrt{L}} \leq 8 \sqrt{2} (1 + 1 / p) < 16.

Dividing the preceding counting inequality by $L$ therefore gives

1 \leq 16 \frac{| J |}{\sqrt{L}} + 4 \frac{X}{L} \leq 16 \sum_{n \in J} (\frac{1}{\sqrt{u_{n}}} + \frac{d_{n} - 2}{u_{n}}),

since $u_{n} < L$ for $n < τ$ and $d_{n} \geq 3$ on $J$ . Thus an arbitrarily late finite window contributes at least $1 / 16$ to $E$ , contradicting convergence. No disjointness of the windows is needed: every window lies in a tail of the nonnegative series. Under the eventual Sylvester recurrence, $u_{n} = 1$ eventually, so there are no late records and $E$ is finite. Finally, term by term at a record,

1_{d_{n} \geq 3} u_{n}^{- 1 / 2} + (d_{n} - 2)_{+} u_{n}^{- 1} \leq 2 (d_{n} - 2)_{+} u_{n}^{- 1 / 2} .

Hence finiteness of the second series implies finiteness of the first, and convergence in the converse direction again follows from eventual Sylvester behaviour. ◻

Formalisation.

The plectis-erdos-lean repository contains the integer inequality counting crossings before a prime power is lost and its counting lemmas. In Lean, the two-unit record-rise implication assumes that arbitrarily late $s$ admit an odd $p^{k} ∣ v_{s}$ with $3 R_{s} < p^{k}$ ; Lemma 7.7 supplies such prime powers.

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.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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