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

Fractions in lowest terms.

Put

G_{n} = gcd (C_{n}, D_{n}), u_{n} = C_{n} / G_{n}, v_{n} = D_{n} / G_{n}, {\tilde{e}}_{n} = E_{n} / G_{n} .

Thus $gcd (u_{n}, v_{n}) = 1$ and ${\tilde{e}}_{n}$ is a signed integer. The factor $h_{n} = G_{n + 1} / G_{n}$ records the common factor removed at the next step. The relation to the preceding section is exact:

\frac{G_{n}}{M_{n}} = gcd (U_{n}, L_{n}), u_{n} = \frac{U_{n}}{gcd (U_{n}, L_{n})}, v_{n} = \frac{L_{n}}{gcd (U_{n}, L_{n})} .

Indeed, $C_{n} = M_{n} U_{n}$ and $D_{n} = M_{n} L_{n}$ . Clearing with an LCM and reducing to lowest terms are different operations. For example, the exact step $(C, D, a) = (3, 6, 3)$ has $E = 0$ and gives $(C^{'}, D^{'}) = (3, 18)$ . With $L = 6$ , its LCM numerator falls from $3$ to $1$ , while the reduced numerator stays $1$ : the LCM factor is $ρ = 3$ , but the reduction factor is $h = 1$ . From this step onwards, the Sylvester recurrence holds.

In general, before reduction the next numerator is $w_{n} = a_{n} u_{n} - v_{n} = u_{n} - {\tilde{e}}_{n}$ , so

h_{n} u_{n + 1} = w_{n}, h_{n} v_{n + 1} = a_{n} v_{n}, gcd (u_{n}, u_{n + 1}) = 1.

The negative magnitude $e_{n} = - E_{n}$ used earlier is not ${\tilde{e}}_{n}$ ; neither is Koizumi’s real gap $ε_{n}$ .

For the maxima and their increments write

R_{n} = max_{k \leq n} u_{k}, H_{n} = max_{j \leq n} C_{j}, s_{n} = R_{n + 1} - R_{n} .

At a strict rise let $d_{n} = u_{n + 1} - u_{n}$ . Then $s_{n} = (d_{n} - (R_{n} - u_{n}))_{+}$ . Thus the actual jump includes recovery of the earlier decrease $R_{n} - u_{n}$ , whereas the record increment does not. For example, if $R_{n} = 10$ , $u_{n} = 5$ and $u_{n + 1} = 12$ , then $s_{n} = 2$ but $d_{n} = 7$ . This illustrates the definitions, not a claimed reciprocal-tail orbit. We will use

m_{n} = (- {\tilde{e}}_{n})_{+}, A_{n} = \frac{R_{n}}{u_{n}} m_{n}, δ_{n} = {(\frac{a_{n}^{2}}{a_{n + 1}} - 1)}_{+}, ℓ (x) = \log_{2} \log_{2} max (4, x) .

The factor $R_{n} / u_{n}$ in $A_{n}$ measures how far the current numerator has fallen below its previous maximum. It equals one at a maximum and can be large after a decrease. The symbol $A_{n}$ is distinct from the prefix product $A_{n}$ used later.

Unless a statement in this section specifies an abstract recurrence, we assume the hypotheses of Problem 1.1 and use its integer tails, for which $| E_{n} | / C_{n} \to 0$ after a finite shift. We will write out the conclusion $a_{n + 1} = a_{n}^{2} - a_{n} + 1$ eventually rather than give it a separate symbol.

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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