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

Unproved inputs for the recorded implications

The following estimates would supply inputs to particular conditional arguments. We compare each with the bounds and finite calculations already available, retaining its quantifiers and evidence basis. The comparisons leave both rational targets and the universal problem open, and do not supply a single obstruction common to all approaches.

For example, the first two weighted divisor counts after $M = 3$ give $(τ (4) - 1) / 2 + (τ (5) - 1) / 4 = 5 / 4$ . The finite weighted divisor sum below records the same calculation for any starting index and length. Its role in particular universal-support criteria is discussed separately; it is not a proved common necessary obstruction for the two problems.

Definition 11.1 (The short-window divisor phase). For $M \geq 1$ and $L \geq 1$ put

Θ_{L} (M) := \sum_{i = 1}^{L} (τ (M + i) - 1) 2^{- i} \in Q_{\geq 0},

where $τ$ is the number-of-divisors function, so that $τ (n) - 1 = # {d \geq 2 : d ∣ n}$ counts the divisors of $n$ other than $1$ .

Iteration of the integer recurrence $K (M + 1) = 2 K (M) - (τ (M + 1) - 1)$ gives the exact identity

Θ_{L} (M) = K (M) - 2^{- L} K (M + L) (M \geq 2, L \geq 1) .

Consequently $2^{L} Θ_{L} (M)$ is an integer, and

dist (Θ_{L} (M), Z) = 2^{- L} dist (K (M + L), 2^{L} Z) .

A nonzero distance is at least $2^{- L}$ , but nonvanishing itself requires an argument. For instance $Θ_{1} (3) = 1$ has distance zero. The identity concerns $K$ ; the quotient deviation also contains the skip-set correction in Theorem 5.14. No positive lower bound for this phase is asserted to be necessary for the universal support problem.

Reset bounds and finite weighted divisor sums

Progress to larger endpoint depths

Prime support and the limits of one formal argument

Equivalent carry conditions at perfect-square depths

Sparse and dense supports under the certificate criterion

What the experiments and obstructions do not imply

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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