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

Centred carries at skipped ranks

Theorem 6.115 (Nonnegative centred carries below the half-value). If $1 \notin A$ and $X_{A} (2) < 1 / 2$ , then

C_{A} (N) = ihc (A, N) - 1 \geq 0 (N \geq 0) .

Indeed, Lemma 2.4 gives

ihc (A, N) = 2^{N + 1} (1 / 2 - X_{A} (2)) + \sum_{r \geq 1} c_{A} (N + 1 + r) 2^{- r} > 0.

The first term is strictly positive and the tail is nonnegative. Since the half-carry is an integer, it is at least $1$ ; subtracting $1$ proves the centred bound. Nonnegativity alone, without this strictness and integrality step, would only give $C_{A} (N) \geq - 1$ . The argument supplies the lower bound used with the conditional upper bound in Theorem 6.84.

Theorem 6.116 (Three possibilities at a skipped endpoint). For a rank $s \geq 5$ omitted by the real greedy support, let $H_{s}$ and $f_{s}$ be as in Lemma 6.45. That lemma identifies the actual prefix as $D_{s}$ when $f_{s} \leq 0$ and as $B_{s}$ when $f_{s} > 0$ , with the exact remainder or overshoot in each case. To obtain the nonnegative-margin condition in Definition 6.81, the negative case would still have to be excluded at every required skipped rank. The trichotomy itself does not exclude it. In particular, the value $- 3$ excluded for $C_{D}$ under the all-right-tail hypothesis of Theorem 6.44 is not an exclusion for $f_{s}$ : the coordinates and the hypotheses are different.

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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