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

Implication table

Read left to right: $\Rightarrow$ records a proved implication with the hypotheses stated in the relevant section, and $\Leftrightarrow$ records an equivalence. Lean proofs and ordinary arguments are distinguished as in the text. “No implication stated” means that no comparison is established here. Quantitative bounds are not ranked solely by their widths.

Target	Relation to its endpoint	Status
Universal irrationality	Would exclude both rational targets; the stronger universal certificate condition is false	open
Half-membership	Would give a counterexample to universal irrationality	open
$1 / 21$ membership	Equivalent to excluding the specified failure branch; would give a counterexample	open
Infinitely many greedy skips (Proposition 12.3)	$\Leftrightarrow$ half-membership	open, equivalent condition
Terminal-only strip (Prop. 12.4)	$\Leftrightarrow$ half-membership	open, equivalent condition
The centred square-root bound for $G$	$\Leftrightarrow$ half-membership	open, equivalent condition
Exact finite sums at unbounded depths	$\Leftrightarrow$ half-membership (Prop. 2.7(c)); Lean proves $\Rightarrow$	open, equivalent condition
Linear-width capacity band	$\Rightarrow$ exact sums at unbounded depths (Lean, given a lemma)	open
The quotient condition on skipped greedy prefixes	$\Rightarrow$ exact sums at unbounded depths (Lean)	open; no strict comparison asserted
Reset square-root bound	$\Rightarrow$ half-membership (Lean); no run-length equivalence asserted	open
The reset sign condition	no implication to half-membership stated	open
Rigidity after long runs	partial rigidity, no exclusion	open
The remaining middle-cell exclusions and right-branch bound	$\Rightarrow$ two-sided bound (Lean, given the hypothesis)	open
The upper-reset band condition	$\Rightarrow$ half-membership (Lean); linear width	open; verified for $13 \leq r \leq 30$

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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