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

Reducing the dyadic-boundary checks

Theorem 6.79 (Testing one nearest dyadic boundary). Let $d, E$ be nonnegative integers with $E \leq 2^{d + 1}$ , and let

j_{*} := max {0 \leq j \leq d : E \leq 2^{d - j + 1}} .

The boundary $2^{d - j_{*} + 1}$ is the smallest power at least $E$ among the finite list $2, 4, \dots, 2^{d + 1}$ ; in particular $j_{*} = d$ when $E \leq 2$ . The following conditions are equivalent:

\begin{matrix} for every 0 \leq j \leq d, 2^{d - j + 1} < E or E + 2 (d + j) \leq 2^{d - j + 1}; \\ E + 2 (d + j_{*}) \leq 2^{d - j_{*} + 1} . \end{matrix}

Proof. For $j > j_{*}$ the boundary is below $E$ ; for $j < j_{*}$ the boundary is larger and the required width $2 (d + j)$ is smaller. This proves sufficiency of the single inequality; necessity is its instance at $j_{*}$ . ◻

For example, $d = 5$ , $E = 20$ gives $j_{*} = 1$ and $20 + 12 = 32$ , so equality at the allowed edge satisfies every band condition.

At an actual upper reset, Proposition 6.80 supplies $E \leq 2^{d + 1}$ . The equivalence replaces all its local band tests by one test. The arithmetic inequality at every required reset remains an unproved hypothesis of the membership implication.

Proposition 6.80 (An upper bound for the reset expression). Let $d \geq 5$ be an actual upper-reset row. Write $E_{d}$ for its reset charge, the sum of four times the adjacent upper overshoot and the corresponding nonnegative correction term. The upper-reset identity is

rem (d + 1) + E_{d} = 2^{d + 1} .

The nonnegativity of the successor remainder therefore gives $E_{d} \leq 2^{d + 1}$ , the side condition needed for Theorem 6.79. This argument uses the upper-reset branch assumption. It does not require the separate, conditional two-sided bound of Theorem 6.9, and does not assert the charge bound at arbitrary rows.

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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