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

Removing one case from a sufficient tail condition

Five Lean declarations reduce the cases that remain to be checked in the square-root lower bound on the producer carry at late middle rows (discussed in Part IV), using the proved exclusion of the $- 3$ cell. No strict separation of the existence conditions is inferred from dropping that case.

Definition 6.144. The middle tail condition is the hypothesis that at every genuine middle transition ( $s \geq 13$ , no successor carry, below the terminal-weight threshold) the terminal-augmented binary coefficient tail is strictly below the signed carry (SeamMiddleProducerTailEscape). The reduced middle tail condition is the same hypothesis restricted to cells $\neq - 3$ (SeamMiddleProducerTailEscapeExceptNegThree).

The middle tail condition trivially implies the reduced one (SeamMiddleProducerTailEscape.toExceptNegThree); the substance is in the converse direction. The reduced condition, with cell $- 3$ discharged by the already proved exclusion (middleProducer_neg_three_not_last), already suffices for half-membership:

reduced middle tail condition ⟹ \frac{1}{2} \in A

(half_mem_mersenneAchievementSet_of_middleProducerTailEscapeExceptNegThree). The unreduced condition and the square-root bound both imply it (half_mem_mersenneAchievementSet_of_middleProducerTailEscape). So the square-root bound implies the middle tail condition, which implies the reduced condition, which implies $1 / 2 \in A$ : a chain of sufficient implications. The arrows do not prove that any implication is strict; all three antecedents remain unproved.

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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