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.

Start here, by the editorial route

In this paper

Logarithmic cost under arithmetic sampling

A logarithmic divisor majorant is too weak to give the uniform arithmetic-progression estimate suggested by the fractional-cover proof. We construct a counterexample already visible on complete periods, so an improved incomplete-period error cannot remove it. We then prove an initial-interval estimate. This distinguishes the available logarithmic bound from a converse to the short paper’s divisor-cube estimates.

The ordinary proofs originate in an AI-assisted note of 17 September 2026. The margin gives the three corresponding formal results and their separate comparison records. The ordinary proofs have not had independent human review. For the underlying Lambert subseries, Kovač–Tao treat the same fixed-base object in [19]; their strict-tail argument gives distinct support coding and Cantor topology. Van Doorn–Kovač treat a freely chosen denominator sequence. Their convention is $n_{i + 1} / n_{i} \geq λ$ at every index. For each $1 < λ < 2$ , they construct such a sequence whose finite reciprocal sums include every rational in $[0, 2]$ ; no sequence with $n_{i + 1} / n_{i} \geq 2$ can do this on any nonempty open interval [20]. This is a statement about the rational points of an interval, not an interval consisting entirely of finite sums; there are only countably many finite subsets. Those results provide context, not an arithmetic logarithmic estimate or a solution of fixed-base rational membership. The proof dependencies below are positive divisor majorisation, elementary prime selection, the Chinese remainder theorem, and the original kernel estimate (13). The short paper’s logarithmic lower bound and its bound for $F (q, P)$ concern the same fractional-cover cost; their structured upper bound is not a uniform logarithmic converse.

The two finite costs

The arithmetic lower bound for every fractional cover

What survives without arithmetic restriction

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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