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

Possible inputs and their scope

The following comparisons identify missing hypotheses in the recorded arguments. They neither predict the length of a proof nor classify all possible methods.

The universal statement. The question is $U 257$ itself. The stronger schema $O 1^{'} = \forall A Cert (A)$ is a rejected sufficient strengthening, not future work: Proposition 11.4 already exhibits a squarefree support on which neither the digitwise nor the carry-aware block certificate exists, while Duverney–Tachiya still prove that squarefree value irrational. Supplying $Cert (A)$ for arbitrary infinite supports is therefore not an open equivalent normal form of $U 257$ . The remaining work is the universal irrationality statement, or a different method that covers supports the certificate schemas cannot.

Half-membership, exact sums and middle cells. The sharp-capacity gap $(*) \to (* *)$ and the $- 2, - 1$ middle-cell exclusion are both, at bottom, requests for anti-concentration of an explicit divisor-count quantity ( $S$ , or $4 \cdot rem - belowPulse - 4$ ) away from specified short integer windows. A congruence argument, a quantitative distribution estimate, or another method would have to supply the stated inequalities. The reformulations do not decide which kind of argument can do so.

Reset separation. The specified quotient-row argument needs $| rem (r + 1) - 2^{r + 1} | > 2^{(r + 5) / 2}$ at its reset rows. A weighted divisor window is one proposed source of information. After division by $2^{r + 1}$ , the displayed condition becomes $| λ_{r + 1} | > 2^{(3 - r) / 2}$ . A bound for $dist (Θ_{L} (M), Z)$ concerns a different quantity; Section 11.1 displays the additional skip-set term that a transfer argument would need to control. No such implication or uniform estimate is supplied. Part IV compares this with a totient-window hypothesis for a particular argument about Problem 249. The analogy does not identify the two statements, transfer a theorem between them, or show that either hypothesis is necessary for its original problem.

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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