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.

Start here, by the editorial route

In this paper

The arithmetic lower bound for every fractional cover

Corollary 13.2 (finite-functional separation). For every finite nonempty $F$ , with $Q = lcm (F)$ ,

\begin{matrix} (20) & K_{*} (F) \geq max_{ℓ ∣ Q} P (U_{F} (N) > 1 ∣ ℓ ∣ N), \end{matrix}

where $N$ is uniform modulo $Q$ . The supports of Theorem 13.1 satisfy $K_{*} (F) \geq 1 - e^{- 1}$ and $κ_{1} (F; 1) \leq 30 \log 2 / H$ . In particular, no absolute $C$ gives $K_{*} (F) \leq C κ_{1} (F; 1)$ for all finite nonempty $F$ , even after all covering sets, exponents, coefficients and positive weights have been optimised.

Proof. Fix one finite-cost cover in (18), and write $U_{j} = U_{F_{j}}$ and $B_{j} = 2^{α_{j}}$ . Coverage gives $U_{F} \leq \sum_{j} U_{j}$ . If $U_{F} > 1$ , then $U_{j} > η_{j}$ for at least one index. Subadditivity and the divisor majorants therefore give the pointwise estimate

1_{{U_{F} > 1}} \leq \sum_{j} η_{j}^{- α_{j}} U_{j}^{α_{j}} \leq \sum_{j, d} η_{j}^{- α_{j}} c_{j, d} w_{B_{j}, d} .

Nonnegative interchange and the original geometric-kernel estimate (13), with its factor $1 + 4 L / M$ , imply

D_{L; R, M} 1_{{U_{F} > 1}} \leq (1 + 4 L / M) K .

Keep the cover and $L$ fixed. For each fixed $M$ , let $R \to \infty$ : periodicity of $1_{{U_{F} > 1}}$ gives $P_{L} (U_{F} > 1) \leq (1 + 4 L / M) K$ . Now let $M \to \infty$ to remove the incomplete-period factor, giving $P_{L} (U_{F} > 1) \leq K$ . Only the indicator for the finite set $F$ needs a period; no common period for the covering sets has been assumed. For $L = ℓ ∣ Q$ the resulting probability is the one in the statement. Since this bound holds for each admissible cover and each such $ℓ$ , take the infimum over covers and the maximum over the finitely many choices of $ℓ$ . Finally use the constructed $L ∣ Q$ and let $H$ tend to infinity. ◻

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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