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

Rewriting the signed carry recurrence

Numeral-adjacent boundary words have more structure than the previously used coefficient bound. If the lower suffix numeral is one larger than the upper suffix numeral, there is a unique carry pivot $d$ : rank $d$ enters the lower support, every rank $e$ with $d < e \leq N$ flips from lower-unselected to upper-selected, and all ranks below $d$ agree (exists_carry_pivot_of_wordSuffixNumeral_succ). Hence for every positive coefficient row $m \geq 1$ the boundary discrepancy is the exact divisor-incidence identity

c_{l o w} (m) + # {e : d < e \leq N, e ∣ m} = c_{u p} (m) + 1_{d ∣ m},

not merely an inequality (supportCoeff_add_deepDivisorCount_eq_of_carry_pivot).

The stage-level theorem consumes adjacent profiled prefixes directly (profiledGapStage_carry_pivot_of_hasAdjacentPrefixes). This is a local normal form for the two-sheet boundary. It does not produce adjacent stages at unbounded depths and therefore decides neither half-membership nor universal #257.

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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