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

The greedy rule for one half

Fix the target $1 / 2$ and the weights $x_{n} = 1 / (2^{n} - 1)$ . Here $x_{1} = 1$ , so exponent one cannot occur in a representation of $1 / 2$ by positive summands; the half-target discussion starts at exponent two. (The reciprocal formula is undefined only at exponent zero.) The greedy rule is deterministic. Strict tail domination makes it recover the support of any represented target (Theorem 5.10):

Definition 5.11 (The half-value greedy orbit). Let $ρ_{k} = r_{k - 1} (1 / 2)$ be the residual entering rank $k$ , so $ρ_{2} = 1 / 2$ . In this subsection $x_{k} = w_{k}$ and $T_{k + 1} = R_{k} = \sum_{j > k} w_{j}$ . At rank $k \geq 2$ :

Take iff $ρ_{k} \geq x_{k}$ ; if taken, $ρ_{k + 1} := ρ_{k} - x_{k}$ .
Skip otherwise; $ρ_{k + 1} := ρ_{k}$ .

A skip at rank $k$ is:

safe iff $ρ_{k} \leq T_{k + 1} := \sum_{j > k} x_{j}$ (the total remaining mass does not yet rule out a representation; this inequality alone does not guarantee one);
fatal iff $ρ_{k} \in (T_{k + 1}, x_{k})$ (the residual is too large for the remaining tail to ever reach, yet too small to have been taken at $k$ – an impossible gap).

For example, ranks $2$ and $3$ are selected, leaving $1 / 2 - 1 / 3 - 1 / 7 = 1 / 42$ . Ranks $4$ and $5$ are skipped, while rank $6$ is selected and leaves $1 / 126$ . These decisions illustrate the rule; they do not establish its infinite survival.

By Theorem 5.10, $1 / 2 \in A$ iff no rank of this orbit is ever fatal. Two boundary degeneracies are excluded unconditionally and independently of any open hypothesis:

Remark 5.12 (No-ties: both greedy-orbit boundaries are strict). (a) $ρ_{k} \neq x_{k}$ at any rank: a finite Mersenne sum has odd reduced denominator (each $2^{n} - 1$ is odd), so the residual $1 / 2$ minus a finite prefix cannot equal the odd-denominator number $x_{k}$ , sharpening the general denominator-parity fact used in Observation 5.3. (b) $ρ_{k} \neq T_{k + 1}$ at any rank: the tail $T_{k + 1} = E - 1 - \sum_{2 \leq j \leq k} x_{j}$ , where $E := \sum_{n \geq 1} x_{n}$ is the Erdős–Borwein constant, is irrational by Erdős’s own 1948 theorem (the base- $2$ case of Theorem 5.4), while $ρ_{k}$ is always rational; equality is impossible. Conclusion: both greedy-orbit boundaries are provably strict at every rank, so the orbit’s decisions stabilize after finitely many strict inequalities at any fixed truncation depth.

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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