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

Filling the remaining binary positions

Theorem 6.75 (Filling the remaining binary positions). For any $c \geq 4$ and any finite $D \subseteq [2, c)$ with $value (D) < 1 / 2$ and the sharp capacity

S (D, 1, 2 c - 2) < 2^{c - 2},

there is an exact row $E$ at endpoint $2 c - 2$ with $D \subseteq E$ and every new rank strictly above $c$ . The implication requires the displayed capacity bound, but neither a crossing condition nor a specified real greedy prefix. The crossing hypothesis used in Theorem 6.70 is one way to seek that input, not an additional premise of this filling result. No strict comparison between the corresponding cofinal existence statements is asserted.

Theorem 6.76 (An exact sum at depth $2 c - 2$ ). Under the hypotheses of Theorem 6.75, there is a set $E \subseteq {2, \dots, 2 c - 2}$ with

Q (E, 2 c - 2) = 2^{2 c - 3} - 1.

This conclusion forgets the additional support-extension information in that theorem. It is the form used when only the existence of an exact row is needed.

Proposition 6.77 (Quotients in the upper half of the index range). For integers $d \geq 2$ and $d \leq M < 2 d$ ,

q (M, d) = ⌊ \frac{2^{M}}{2^{d} - 1} ⌋ = 2^{M - d} .

Indeed, the quotient is $2^{M - d} + 2^{M - d} / (2^{d} - 1)$ , whose second term is strictly between zero and one. In Theorem 6.75, the available ranks $d = c + 1, \dots, 2 c - 2$ therefore supply the binary weights $2^{c - 3}, \dots, 1$ . They represent every integer from $0$ to $2^{c - 2} - 1$ , which explains both the capacity bound and the choice of terminal depth.

Theorem 6.78 (A finite sum from a skipped prefix). Let $c \geq 4$ and $D \subseteq {2, \dots, c - 1}$ satisfy

0 < \frac{1}{2} - X_{D} (2) < \frac{1}{2^{c} - 1} .

Then there is $E$ with $D \subseteq E \subseteq {2, \dots, 2 c - 2}$ and $Q (E, 2 c - 2) = 2^{2 c - 3} - 1$ . No separate sharp-capacity assumption is required. The binary completion in the linked proof uses ranks $c, \dots, 2 c - 2$ ; it may therefore insert $c$ . Theorem 6.75 instead assumes the sharper $(c - 2)$ -bit bound so that every added rank is strictly greater than $c$ .

For a finite $F \subseteq {2, \dots, n}$ with $X_{F} (2) > 1 / 2$ , take $c$ to be its first crossing rank and $D = F \cap {2, \dots, c - 1}$ . Then $c \geq 4$ because $1 / 3 + 1 / 7 < 1 / 2$ , and the displayed hypotheses hold: equality at the lower endpoint is excluded by odd denominators. The Lean proofs use the theorem in exactly this first-crossing form.

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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