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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Constant negative errors

A constant negative error makes the numerator grow linearly. Such behaviour can persist for arbitrarily long finite intervals, even on genuine rational orbits, by the flat-transient theorem in the working report on Problem #243 [9]. We exclude infinite constant and periodic negative tails below. These exclusions supply no bound for the finite transient lengths.

Suppose the error is negative with a constant magnitude, $E_{n} = - m$ for a fixed $m > 0$ and every $n$ . By Proposition 4.1 the numerator increases by $m$ at each step, so $C_{n} = c + n m$ with $c = C_{0}$ , and the definition of the error becomes the identity

\begin{matrix} (5.1) & D_{n} + m = (a_{n} - 1) (c + n m) . \end{matrix}

Together with $D_{n + 1} = a_{n} D_{n}$ this is a closed system in $(a, D)$ . We shall prove that it has no infinite natural-number solution with $a_{n} \geq 2$ , after examining a finite example.

Example 9.1 (a nonintegral second step). Take $m = c = 1$ , so that $C_{n} = 1 + n$ and (5.1) reads $D_{n} + 1 = (a_{n} - 1) (1 + n)$ , and take $D_{0} = 1$ . Each step is now determined: at $n = 0$ the equation reads $2 = (a_{0} - 1) \cdot 1$ , so $a_{0} = 3$ , and $D_{1} = a_{0} D_{0} = 3$ ; at $n = 1$ it reads $4 = (a_{1} - 1) \cdot 2$ , so $a_{1} = 3$ , and $D_{2} = a_{1} D_{1} = 9$ ; at $n = 2$ it reads $10 = (a_{2} - 1) \cdot 3$ , which has no integer solution, and the orbit stops. Longer prefixes occur for other starting values. For $2 \leq a_{0} < 5000$ , the exact residue search in Appendix B gives a maximum of $17$ successful updates, hence $18$ multiplier values including $a_{0}$ .

Theorem 9.2 (no constant negative magnitude). For any $m, c \in N$ with $m > 0$ , there is no pair of sequences $a, D : N \to N$ with $a_{n} \geq 2$ for all $n$ satisfying $D_{n + 1} = a_{n} D_{n}$ and (5.1). The same holds if the identity is assumed only from some index onwards.

Proof when $c$ and $m$ are coprime. Assume first $gcd (c, m) = 1$ . Every multiplier must share a prime with $m$ , but each such prime can occur in at most one multiplier.

Every $a_{j}$ shares a prime with $m$ . Suppose $gcd (a_{j}, m) = 1$ . Then $m$ is invertible modulo $a_{j}$ , so some $n$ has $c + n m \equiv 0$ , that is $a_{j} ∣ C_{n}$ ; and $a_{j} ∣ D_{n}$ for every $n > j$ , since $D$ is multiplicative with $a_{j}$ among its factors. Choosing such an $n$ beyond $j$ and reading (5.1) modulo $a_{j}$ gives $a_{j} ∣ m$ , contradicting $gcd (a_{j}, m) = 1$ and $a_{j} \geq 2$ .

A prime divisor of $m$ occurs in at most one $a_{j}$ . Suppose $p ∣ m$ and $p ∣ a_{j}$ . For $n > j$ we have $p ∣ D_{n}$ , so (5.1) gives $(a_{n} - 1) C_{n} \equiv 0 (\mod p)$ . Now $C_{n} = c + n m \equiv c$ , and $p ∤ c$ because $p ∣ m$ and $gcd (c, m) = 1$ ; hence $p ∣ a_{n} - 1$ , so $p ∤ a_{n}$ . Thus $p$ cannot divide any later multiplier.

This injects infinitely many multipliers into the finite set of prime divisors of $m$ , a contradiction. ◻

The case $gcd (c, m) = 1$ is the exclusion when $c$ and $m$ are coprime. The special case $m = c = 1$ , worked through in Example 9.1, where $m$ has no prime divisors at all and the first fact is immediately contradictory, is recorded separately as the case $c = m = 1$ .

Removing the scale. For general $c$ , set $g = gcd (c, m)$ . Equation (5.1) gives $g ∣ D_{n}$ . Both identities are homogeneous in $D, c, m$ : dividing them by $g$ leaves the multipliers unchanged and gives coprime data, excluded by the previous case. ◻

The formal statements are the exclusion at every scale and the eventual exclusion. The latter follows by shifting the orbit to the first index of constant error.

Theorem 9.2 excludes an error that is eventually constant and negative, at every magnitude and every scale. Constancy fixes both the possible prime divisors and the congruence used at later indices. For varying magnitudes, even when their prime divisors belong to a fixed finite set, this argument does not supply the same later congruence. It therefore does not settle that case.

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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