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

Compatible finite approximations

Theorem 6.69 (A compatible family of finite supports). Let $b_{n, d} \in {0, 1}$ satisfy $b_{n + 1, d} = b_{n, d}$ whenever $2 d \leq n$ . Define

E_{n} = {d : 2 \leq d \leq n, b_{n, d} = 1}, D_{n} = E_{n} \cap {2, \dots, ⌊ n / 2 ⌋} .

The four conditions on the finite rows, required for every $n \geq 2$ , are

\begin{aligned} 2^{max {c_{D_{n}} (n) - 1, 0}} - 1 & \leq S (D_{n}, 1, n - 1), \\ \sum_{\begin{matrix} d \in E_{n} \\ d > ⌊ n / 2 ⌋ \end{matrix}} 2^{n - d} & = H (D_{n}, 1, n), \\ H (D_{n}, 1, n) & < 2^{n - ⌊ n / 2 ⌋}, \\ Q (E_{n}, n) & = 2^{n - 1} - 1. \end{aligned}

The first condition implies that the integer in the next two lines is nonnegative. The second specifies the binary value of the upper half of the row, the third bounds that value by its available number of bits, and the fourth is the exact quotient equation. The maximum in the first line records natural-number subtraction at the case $c_{D_{n}} (n) = 0$ .

These are the finite hypotheses of the compatible-limit construction. The bit condition fixes coordinate $d$ from row $2 d$ onward. Unlike the independent finite supports in Definition 6.66, the rows therefore have a prescribed common limit. This extra structure is not, by itself, a proof of strictness between the corresponding existence statements.

The existence of rows satisfying these four conditions for every $n \geq 2$ is not proved.

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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