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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Bounded negative part

The lower bound $E_{n} \geq - B$ has two uses. At negative indices it bounds the gcd, making it stable. At every index it bounds the upward step $C_{n + 1} - C_{n} = - E_{n}$ . We can therefore combine the preceding coprimality and first-crossing arguments.

Without the denominator recurrence, $C_{n} = c + b n$ , $E_{n} = - b$ would be a counterexample for any positive integers $b, c$ . Theorem 9.2 excludes this exact-orbit behaviour. The separate summability criterion in the next section needs only the scalar update. For positive integer solutions of (19) with vanishing relative error, both criteria are equivalent to eventual zero error. Neither extra estimate has been derived from the unrestricted hypotheses.

Theorem 12.1 (bounded negative part). Let $a, C, D : N \to N$ satisfy $a_{n} > 1$ , $C_{n} > 0$ , and

C_{n + 1} + D_{n} = a_{n} C_{n}, D_{n + 1} = a_{n} D_{n} (n \geq 0) .

Put $E_{n} = D_{n} - (a_{n} - 1) C_{n}$ . If

\frac{E_{n}}{C_{n}} ⟶ 0 and E_{n} \geq - B for all sufficiently large n

for some integer $B \geq 0$ , then $E_{n} = 0$ for all sufficiently large $n$ .

The limit is equivalent, for positive $C_{n}$ , to the integer condition $K | E_{n} | < C_{n}$ eventually for each $K \in N$ . This is the form used by the linked declaration. Its $K = 1$ case gives $| E_{n} | < C_{n}$ eventually.

Proof. Suppose that the error fails to vanish eventually. After the point from which $| E_{n} | < C_{n}$ always holds, Theorem 5.8 excludes every zero. Hence $| E_{n} | \geq 1$ , and the relative limit forces $C_{n} \to \infty$ .

There are infinitely many negative indices, since otherwise Theorem 8.1 would give eventual zero. At each sufficiently late negative index, $1 \leq - E_{n} \leq B$ , so $B \geq 1$ . Proposition 11.6 makes $G_{n} = gcd (C_{n}, D_{n})$ stabilise to an integer $g > 0$ . The divided sequences $u_{n} = C_{n} / g$ , $v_{n} = D_{n} / g$ satisfy the coprime recurrences, with $u_{n} \to \infty$ and $u_{n + 1} - u_{n} = - E_{n} / g \leq B$ . Their coprimality properties from Proposition 11.4 now contradict Theorem 11.3. ◻

Corollary 12.2. Under the hypotheses of Theorem 12.1, the multipliers satisfy $a_{n + 1} = a_{n}^{2} - a_{n} + 1$ for all sufficiently large $n$ .

Proof. Theorem 12.1 gives eventual zero error. Since $C_{n + 1} > 0$ , Theorem 5.7 gives the recurrence. ◻

To apply it to Problem 1.1, delete a finite prefix and use Koizumi’s Corollary 3 [12]. The remaining sequence is the pseudo-greedy expansion of its own reciprocal sum. His Lemma 4 gives positive integer numerators $c_{n}$ , denominators $d_{n}$ and errors $e_{n}$ with

c_{n + 1} = a_{n} c_{n} - d_{n}, d_{n + 1} = a_{n} d_{n}, e_{n} = d_{n} - (a_{n} - 1) c_{n} .

The terms $a_{n}$ eventually exceed one, and $e_{n} / c_{n} \to 0$ by that corollary. The same construction gives $- c_{n} / 2 \leq e_{n} < c_{n} / 2$ . The additional condition still required is an eventual lower bound $e_{n} \geq - B$ . Neither Koizumi’s construction nor the constant and periodic exclusions above establish this bound for every rational sum in the problem. The result therefore remains conditional.

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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