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

Multiplicity of the stated two–three representations

Consider the primitive representations $2 p + 3 q = n$ with $p, q > 0$ and $gcd (p, q) = 1$ (IsPrimitive23Solution). Every $n \geq 11$ has such a representation (exists_primitive23_solution_of_eleven_le), while $n = 10$ has none (no_primitive23_solution_ten). The value $n = 11$ already has two distinct representations (exists_two_primitive23_solutions_eleven), and so does every $n = 10 k$ with $k \geq 2$ (exists_two_primitive23_solutions_mul_ten). These facts explain an approach and give no counterexample: nothing here proves that the associated integer-multiplicity expansion of $1 / 21$ can be made Boolean. The recurring multiplicities are precisely the unresolved collision data, not zero–one support digits.

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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