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

Integer numerators, denominators and errors

We retain an unreduced fraction $C_{n} / D_{n}$ for each rational tail. This keeps the denominator update multiplicative: removing $1 / a_{n}$ from the tail gives

\begin{matrix} (19) & D_{n + 1} = a_{n} D_{n}, C_{n + 1} = a_{n} C_{n} - D_{n} . \end{matrix}

We measure the difference from the Sylvester recurrence by $E_{n} = D_{n} - (a_{n} - 1) C_{n}$ . These are Koizumi’s recurrences [12]; their formal definitions are the denominator update, the numerator update, and the error.

We also consider integer solutions of (19) without initially assuming that $C_{n} / D_{n}$ is a reciprocal tail. Here and in the following recurrence sections the indices may start at $0$ , and $E_{n}$ always denotes the error just defined. We state positivity and growth assumptions as they are needed. The terms $a_{n}$ are the multipliers, while $C_{n}, D_{n}$ are the numerator and denominator. The pseudo-greedy rounding rule is an additional condition giving $- C_{n} / 2 \leq E_{n} < C_{n} / 2$ , whereas the absorption argument below needs only $| E_{n} | < C_{n}$ . At a negative error we write $e_{n} = - E_{n} > 0$ . For an isolated step we use $a, D, C \in Z$ without subscripts.

Proposition 4.1 (update law). For all $a, D, C \in Z$ ,

a C - D = C - (D - (a - 1) C) .

Consequently, every solution of (19) satisfies $C_{n + 1} = C_{n} - E_{n}$ .

Proof. $C - (D - (a - 1) C) = a C - D$ . ◻

$C_{n}$ decreases when $E_{n} > 0$ , increases when $E_{n} < 0$ , and stays unchanged when $E_{n} = 0$ . This is why bounds on the negative error become bounds on upward increments.

Example 4.2 (the Sylvester sequence). Take $a_{n} = 2, 3, 7, 43, 1807, \dots$ with $a_{n + 1} = a_{n}^{2} - a_{n} + 1$ , and start the state at $D_{0} = C_{0} = 1$ . The two updates give

$n$	$a_{n}$	$D_{n}$	$C_{n}$	$E_{n}$
$0$	$2$	$1$	$1$	$0$
$1$	$3$	$2$	$1$	$0$
$2$	$7$	$6$	$1$	$0$
$3$	$43$	$42$	$1$	$0$
$4$	$1807$	$1806$	$1$	$0$

and $D_{n} = a_{n} - 1$ with $C_{n} = 1$ at every index: if $D_{n} = a_{n} - 1$ and $C_{n} = 1$ then $C_{n + 1} = a_{n} - (a_{n} - 1) = 1$ and $D_{n + 1} = a_{n} (a_{n} - 1) = a_{n + 1} - 1$ . Hence $E_{n} = D_{n} - (a_{n} - 1) C_{n} = 0$ throughout and the numerator never moves, which is Proposition 4.1 in the stationary case.

Construction from the reciprocal sum.

Let $T_{n} = \sum_{k \geq n} 1 / a_{k}$ and suppose $T_{0} \in Q$ . Put $D_{0}$ equal to a common denominator and $D_{n} = D_{0} a_{0} \dots a_{n - 1}$ , and set $C_{n} = D_{n} T_{n}$ . Then $C_{n} \in Z$ for every $n$ , and from $T_{n} = 1 / a_{n} + T_{n + 1}$ one gets $C_{n + 1} = a_{n} C_{n} - D_{n}$ , which is the tail update. On Sylvester’s sequence $T_{n} = 1 / (a_{n} - 1)$ exactly, so $D_{n} = (a_{n} - 1) C_{n}$ and $E_{n} = 0$ : the error measures deviation from the telescoping tail identity, and it vanishes identically on the Sylvester orbit.

In Lean, the elementary identities are stated for an abstract integer system, and their application to a rational reciprocal sum has separate statements, including the product condition for a rational reciprocal sum. This passage uses the tail realisation, the denominator realisation and the signed update.

Conversely, the recurrences identify a reciprocal sum when the ratio $C_{n} / D_{n}$ tends to zero. Regard that ratio as a real number, the tail ratio. Assume $a_{n} > 0$ , $D_{n} > 0$ , $C_{n + 1} + D_{n} = a_{n} C_{n}$ and $D_{n + 1} = a_{n} D_{n}$ . Dividing the numerator update by $a_{n} D_{n}$ gives $C_{n} / D_{n} = 1 / a_{n} + C_{n + 1} / D_{n + 1}$ , the one-step reciprocal identity. Iterating gives

\frac{C_{0}}{D_{0}} = \sum_{n < N} \frac{1}{a_{n}} + \frac{C_{N}}{D_{N}},

as in the finite telescoping identity. If $C_{N} / D_{N} \to 0$ , taking the limit identifies $\sum_{n} 1 / a_{n} = C_{0} / D_{0}$ , the series realisation. No growth, error bound or separate rationality hypothesis is used in these three identities. Positivity and the stated limit of $C_{n} / D_{n}$ are still required. They prove the direction from recurrences to a series; the construction from a rational reciprocal sum was given above and is also formalised in separate declarations.

Under the relative-error hypothesis used below, the required limit is automatic. Suppose $a_{n} > 1$ , $C_{n} > 0$ , $D_{n} \geq 0$ and $E_{n} / C_{n} \to 0$ on an exact integer orbit. If $D_{0} = 0$ , then $D_{n} = 0$ and $E_{n} / C_{n} = 1 - a_{n} \leq - 1$ at every index, a contradiction. Thus $D_{0} \geq 1$ and $D_{n} \geq D_{0} 2^{n}$ . For each $ε > 0$ , the update gives $C_{n + 1} \leq (1 + ε) C_{n}$ eventually. Taking $0 < ε < 1$ shows that $C_{n} / D_{n} \to 0$ , so the telescoping identity realises the reciprocal sum as $C_{0} / D_{0}$ .

The same assumptions also imply the quadratic growth limit. Put $θ_{n} = E_{n} / C_{n}$ . Then $a_{n} = D_{n} / C_{n} + 1 - θ_{n} \to \infty$ , and the exact recurrences give

\frac{a_{n + 1}}{a_{n}^{2}} = \frac{1}{1 - θ_{n}} - \frac{1}{a_{n}} + \frac{1 - θ_{n + 1}}{a_{n}^{2}} ⟶ 1.

In particular, the multipliers are strictly increasing eventually. Thus a global orbit with these positivity and relative-error assumptions already gives a sequence of the type in Problem 1.1 after deletion of a finite prefix. No separate tail-limit estimate is needed for such an orbit; without the relative-error assumption, that estimate remains a condition of the general telescoping argument.

Proposition 4.3 (scaling the numerator and denominator). For every $s, a, D, C \in Z$ ,

a (s D) = s (a D), a (s C) - s D = s (a C - D),

s D - (a - 1) s C = s [D - (a - 1) C] .

Multiplying the numerator and denominator by the same factor therefore preserves the recurrence and scales the error by that factor. When the factor divides all entries, we can divide it out instead. This is used in the finite calculation of Appendix B and in the induction in Theorem 10.1.

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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