Reciprocal Mersenne Subseries

Will Cook

Long paper

Reciprocal Mersenne Subseries

Erdős #257 By Will Cook 30 September 2026 Authorship, AI use and citation

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Summary. A weighted summability criterion proves irrationality of reciprocal Mersenne subseries: if h is the part of a supported on a finite nonempty prime set P, every infinite A with ∑ $h (a) / (a (b^{h (a)} - 1))$ finite has irrational ∑ $(b^{a} - 1)^{- 1}$ . The base-two condition gives the conclusion at every integer base for every infinite subset, and permits an example with divergent reciprocal sum, while excluding full support and all odd exponents. The proof averages over a finite part of the support and over dyadic observation lengths. Erdős's weaker reciprocal-summable theorem is proved in full, and a common window extends both arguments to unions of weighted and positive-cover supports. Finite-denominator and achievement-set calculations formulate tests for the targets $1 / 2$ and $1 / 21$ ; those tests do not decide either target or the universal question. Prime support at base 2 and squarefree support at power-of-two bases are attributed as existing work.

This paper covers a claim-bounded problem-specific reasoning surface for Erdős #257: selected checked premises, finite computations, closed routes, surviving obligations, and their exact evidence bands.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #257, whose universal and half-value questions remain open.

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Comparison with the known theorem.

Luca and Tachiya proved irrationality for every nonzero purely periodic integer weight, of any signs [3], so the terminating branch below occurs only for the zero weight. The theorem records the dichotomy reached by the certificate method, together with its one-sided closures.

Theorem 6.23 (A formal dichotomy for signed periodic weights). Let $b \geq 2$ and $m \geq 1$ be integers, and let $w : N_{> 0} \to Z$ be $m$ -periodic. Put

x = \sum_{a \geq 1} \frac{w (a)}{b^{a} - 1}, c_{w} (n) = \sum_{d ∣ n} w (d) .

The cited formal argument gives the dichotomy that $x$ is irrational or $b^{k} x \in Z$ for some integer $k \geq 0$ . It excludes the latter alternative if $c_{w}$ has one sign throughout and is nonzero at arbitrarily large indices. These sign assumptions concern $c_{w}$ , not $w$ . As noted above, Luca and Tachiya’s theorem already excludes the terminating alternative for every nonzero periodic $w$ , without either sign restriction. The zero weight gives $x = 0$ .

For the next two results write $M_{r} = 2^{r} - 1$ and, for a positive squarefree integer $r$ , set

A_{r} = \sum_{d ∣ r} μ (d) \frac{r}{d} \frac{M_{r}}{M_{d}} \in Z, B (r) = \frac{A_{r}}{M_{r}} = \sum_{d ∣ r} \frac{μ (d) (r / d)}{2^{d} - 1} .

Every quotient in $A_{r}$ is integral because $d ∣ r$ . The Lean source calls $A_{r}$ the Möbius numerator; $B (r)$ is a finite rational sum, not the full-support Lambert series.

Theorem 6.24 (Denominators of finite Mersenne sums). Let $t \geq 1$ and $h \geq 1$ be integers, and let $r \geq 1$ be squarefree with every prime factor at most $t$ . Let $P$ be a set of prime divisors of $r$ such that $t < 2 p$ for each $p \in P$ , and put $C = \prod_{p \in P} (2^{p} - 1)$ . Then

\frac{C}{gcd (C, h)} | den (h B (r)) .

In particular, $C$ divides this reduced denominator when $gcd (C, h) = 1$ . The factors in $C$ are pairwise coprime, since $gcd (2^{p} - 1, 2^{q} - 1) = 2^{gcd (p, q)} - 1 = 1$ for distinct primes. The assertion concerns this finite signed sum; it does not replace the approximation hypotheses in Theorem 6.14.

Theorem 6.25 (Exponential growth of the reduced denominators). For an integer $t \geq 5$ , put

H_{t} = lcm (1, \dots, t), r_{t} = \prod_{p \leq t} p, h_{t} = H_{t} / r_{t}, D_{t} = den (h_{t} B (r_{t})),

where the product is over primes. Then $D_{t} \geq 2^{t / 2}$ . Its exact denominator is

D_{t} = \frac{2^{r_{t}} - 1}{gcd (2^{r_{t}} - 1, h_{t} A_{r_{t}})} .

Proof. Apply Bertrand’s postulate [24] to $m = ⌈ t / 2 ⌉ \geq 3$ . There is a prime $m < p < 2 m$ , hence $p \leq t$ and $p - 1 \geq t / 2$ . Every prime divisor $ℓ$ of $2^{p} - 1$ has multiplicative order $p$ for $2$ modulo $ℓ$ . Since $p$ is odd, $2 p ∣ ℓ - 1$ ; thus $ℓ > t$ . All prime factors of $h_{t}$ are at most $t$ , so Theorem 6.24 gives

2^{t / 2} \leq 2^{p - 1} < 2^{p} - 1 \leq D_{t} .

The exact expression for $D_{t}$ is the usual reduction of the fraction $h_{t} A_{r_{t}} / (2^{r_{t}} - 1)$ . ◻

The Lean proof of the bound uses natural-number division $t / 2$ , that is, $⌊ t / 2 ⌋$ . The ceiling choice in the ordinary argument supplies the displayed real-exponent bound also when $t$ is odd. The Lean statement of the exact denominator uses an equivalent formula in its own numerator notation. Neither denominator growth nor that identity supplies an approximation-error estimate.

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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