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

Gap lengths and the measure of their union

Definition 6.117 (The gap after a Mersenne weight). For $n \geq 1$ , let $R_{n} = \sum_{j > n} w_{j}$ and $g_{n} = w_{n} - R_{n} > 0$ . The quantity $g_{n}$ is the length of each gap created at level $n$ , not the sum of all gap lengths created at that level.

mersenneGap mersenneGap_pos

Theorem 6.118 (The unweighted sum of remaining gap lengths). For every $N \geq 0$ , the unweighted sum of gap lengths satisfies

\sum_{n > N} g_{n} \leq \frac{2}{9} 4^{- N} + \frac{3}{7} 8^{- N},

which tends to zero. The estimate follows by summing the per-level bound; it counts one length per level. It does not, by itself, bound the measure of the union of all gaps, because there are $2^{n - 1}$ disjoint translated gaps at level $n$ .

The measure of the union of gaps created after level $N$ is instead

\sum_{n > N} 2^{n - 1} g_{n} = 2^{N} R_{N} - 1 \leq \frac{2}{3} 2^{- N} (N \geq 1) .

Indeed, the finite sum from $N + 1$ to $M$ telescopes to $2^{N} R_{N} - 2^{M} R_{M}$ , and $2^{M} R_{M} \to 1$ . The final bound uses $R_{N} \leq 2^{- N} + (2 / 3) 4^{- N}$ . Disjointness follows from the strict-tail construction of the achievement set. This is an ordinary geometric deduction, not the statement of the preceding linked gap-sum lemma. A small union measure still gives no answer about a prescribed point.

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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