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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In this paper

The universal statement

Definition 12.1 (Universal #257).

U 257 := \forall A \subseteq N_{\geq 1}, A infinite ⟹ Irrational (X_{A} (2)) .

This is the literal statement of Erdős #257; it is open. The sufficient condition used by the certificate method of this record is

Cert (A) := \forall q > 0, \exists N, K, L, C, K \leq L, (\forall r \in [1, K], 2^{r} ∣ c_{A} (N + r)) \land

\sum_{r = K + 1}^{L} c_{A} (N + r) \cdot 2^{L - r} \leq C \land \exists t, 0 < c_{A} (N + L + 1 + t) \land q (C + N + L + 2) < 2^{L},

with $Cert (A) \Rightarrow Irrational (X_{A} (2))$ already proved unconditionally for every $A$ (irrational_erdosSupportSeries_of_weighted_coeff_certificates; the theorem is universally quantified over $A$ , and only its hypothesis is missing). Therefore the stronger statement

O 1^{'} := \forall A, A infinite ⟹ Cert (A) .

would imply $U 257$ , but the reduction is not lossless and $O 1^{'}$ is actually false. For $A = {n \geq 2 : n squarefree}$ the incidence is $2^{ω (n)} - 1$ , hence odd for every $n \geq 2$ , and the digitwise certificate schema has no instance at any even base (ErdosProblems.Erdos257.not_exists_digitwise_certificates_squarefreeSupport). This is a limit of the method, not a counterexample to irrationality: Duverney–Tachiya prove the squarefree value irrational at every power-of-two base. Likewise the pairwise-coprime theorem does not cover the primes because $\sum_{p} 1 / p$ diverges, even though Tao–Teräväinen prove the base- $2$ prime-support value irrational. The universal target is $U 257$ itself; $O 1^{'}$ is a rejected sufficient strengthening, not an open equivalent normal form.

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.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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