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

Hypotheses of the CRT block-certificate argument

Observation 2.13 (Two inputs to the CRT certificate argument). The CRT block-certificate arguments considered here require both compatible divisibility prescriptions and a tail estimate strong enough for the height inequality $q (C + N + L + 2) < b^{L}$ . Pairwise coprimality is one way to obtain the first input; reciprocal summability is one way to obtain the second. This observation concerns those certificate arguments only. It does not describe all theorems in Section 1: the reciprocal-summable and weighted averaging proofs do not require pairwise coprimality.

What remains to check. One cannot apply a conditional CRT certificate theorem to arbitrary infinite $A$ without establishing its hypotheses. Shared factors impose compatibility conditions on the prescribed residues; they do not make every CRT construction impossible. The certificate proof of Theorem 6.15 uses the tail budget in one step. Deleting that assumption does not complete the proof: one needs a replacement estimate for that step. This leaves open different constructions and different averaging arguments.

Explains only a method boundary. The primes are infinite and pairwise coprime, so they satisfy the first input. The identity compositeDilationDefect_eq_zero_of_prime_support makes the dilation algebra exact there; yet this certificate theorem does not apply because $\sum_{p} 1 / p$ diverges, so the stated reciprocal-sum hypothesis fails. This explains the absence of a Lean theorem from this certificate argument; it does not make the base- $2$ value open, since Tao–Teräväinen settle it by a different analytic method. The sunflower generalisation inherits both axes and therefore does not recover that external theorem.

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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