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

Finite approximations without compatibility

For a finite support $D \subseteq {2, \dots, M}$ , the integer

K_{D} (M) = 2^{M - 1} - \sum_{j = 2}^{M} 2^{M - j} # {d \in D : d ∣ j}

is the half-carry at the terminal rank. It is $ihc (D, M - 1)$ in the notation used earlier. The condition below asks for $M_{j} \to \infty$ and finite sets $D_{j}$ with $| K_{D_{j}} (M_{j}) | / 2^{M_{j}} \to 0$ . It does not require the sets to agree or bound the carries at earlier ranks. The estimate

| \sum_{d \in D_{j}} \frac{1}{2^{d} - 1} - \frac{1}{2} | \leq \frac{| K_{D_{j}} (M_{j}) | + 2 \sqrt{M_{j}} + 4}{2^{M_{j}}}

follows by expanding each Mersenne weight. Indeed,

X_{D} (2) - \frac{1}{2} = - 2^{- M} K_{D} (M) + \sum_{n > M} c_{D} (n) 2^{- n} .

Here $c_{D} (n) \leq τ (n) \leq 2 \sqrt{n}$ , and $\sqrt{M + r} \leq \sqrt{M} + r$ gives

\sum_{r \geq 1} 2^{- r} c_{D} (M + r) \leq 2 \sqrt{M} + 4.

Each finite value belongs to the closed achievement set; its limit therefore belongs too, even when the finite supports disagree. No compatibility between them is needed for this implication. The finite approximants themselves remain an unproved input. A square-root terminal bound is sufficient and stronger as a bound on these particular witnesses; it does not establish a strict logical separation between the corresponding existence statements.

Here is the finite family used in the cited construction. Put $B (m) = 2 ⌊ \sqrt{m} ⌋ + 4$ . A depth- $M$ family with a shared prefix through $K \leq M$ consists of sets $A_{1}, \dots, A_{B (M)} \subseteq {2, \dots, M}$ such that

1 \leq K_{A_{k}} (m) \leq B (m) (1 \leq m \leq M), K_{A_{k}} (M) = k,

all $A_{k}$ agree through $K$ , and one integer $E \geq B (M)$ satisfies

\sum_{j = K + 1}^{M} 1_{A_{k}} (j) 2^{M - j} + k = E (1 \leq k \leq B (M)) .

The suffixes therefore have consecutive binary values $E - k$ . The following statements explain how these particular families supply finite approximations; they do not construct them at arbitrary depths.

Proposition 6.142. Every shared-prefix family just defined at depth $N$ contains a finite support $A$ with $| K_{A} (N) | \leq B (N)$ , and at a feedback row this survives both outputs of the corpus’s total feedback theorem; full-cylinder advance or a localized one-hole seam, which can delete at most one of carries $3, 4$ .

Four concrete instances, at depths $51$ – $54$ , are verified by exact computation of the underlying stages (fullCylinderStage51_halfTerminalOnlyStripWitness, fullCylinderStage54_halfTerminalOnlyStripWitness). This is a finite chain of witnesses; it does not reach unbounded depths.

Observation 6.143 (Shared-prefix families at unbounded depths). The conditional theorem

(\begin{matrix} such shared-prefix families exist \\ at arbitrarily large depths M \end{matrix}) ⟹ \exists A infinite with X_{A} (2) = 1 / 2

is proved unconditionally (exists_infinite_support_half_of_cofinalCylinderStages), with a positive-support refinement using the same odd-denominator lemma (finite_boolSupport_ne_half; exists_infinite_positive_support_half_of_cofinalCylinderStages). The hypothesis requires these finite families at arbitrarily large depths and is unproved. The terminal bound then implies half-membership. This is a sufficient condition; no converse or strict comparison with other sufficient conditions is established here.

For the conditional implication, choose the member with terminal carry $1$ from each shared-prefix family. Its error in approximating $1 / 2$ is at most $(1 + 2 \sqrt{M} + 4) 2^{- M}$ by the preceding estimate. At unbounded depths these errors tend to zero, so compactness gives a representation. Every finite sum has odd reduced denominator; hence that representation uses an infinite support. This deduction needs no compatibility between the chosen members of different families. It still assumes that the stated families exist at unbounded depths.

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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