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

A real-valued form of the quotient identity

Theorem 5.15 (Real (non-integer) form of the quotient identity). Let $x_{d} := 1 / (2^{d} - 1)$ and define the constant

C := \sum_{d \geq 2} x_{d} - \frac{1}{2} = E - \frac{3}{2} = 0.1066951524152917 \dots,

where $E$ is the Erdős–Borwein constant. Then, for every quotient row $n \geq 6$ ,

\begin{matrix} (II) & Δ_{n} = 4^{n} (\sum_{d \in {Skip}_{n}} x_{d} - C) + η_{n}, | η_{n} | < 2 n + 2. \end{matrix}

Proof. Splitting the full series at $n$ gives

η_{n} = 4^{n} R_{n - 1} - 2^{n + 1} + \sum_{d \in D_{n}} {\frac{4^{n}}{2^{d} - 1}} .

The tail bounds $2^{- (n - 1)} < R_{n - 1} \leq 2^{- (n - 1)} + \frac{2}{3} 4^{- (n - 1)}$ and $| D_{n} | \leq n - 2$ show $0 < η_{n} < n + 2 / 3 < 2 n + 2$ .

Use of the error bound. For $H > 2 n + 2$ , the condition $| \sum_{d \in {Skip}_{n}} x_{d} - C | > (H + 2 n + 2) / 4^{n}$ is sufficient for $| Δ_{n} | > H$ . Conversely, $| Δ_{n} | > H$ implies $| \sum_{d \in {Skip}_{n}} x_{d} - C | > (H - 2 n - 2) / 4^{n}$ . Thus a square-root-exponential deviation corresponds to a separation of order $2^{- 3 n / 2}$ , with the displayed additive error retained. This is not an exact equivalence after simply discarding that error.
Note: $C = E - 3 / 2$ is Mersenne-specific, but the shape “scaled deviation $=$ $4^{n} \cdot$ (finite skip-sum $-$ target constant) $+ O (n)$ ” is a template, not yet matched to any analogous constant on the #249 side.

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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