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

What is proved

Full-support irrationality, for every base

Theorem 5.4 (Full-support irrationality, unconditional, every base $b \geq 2$ ). For every integer $b \geq 2$ ,

\sum_{k = 0}^{\infty} \frac{1}{b^{k + 1} - 1} is irrational.

Hypotheses: $b \in N$ , $b \geq 2$ . Conclusion: unconditional irrationality of the full-support Erdős–Borwein-type series at every base, not merely base $2$ . The base- $2$ instance recovering the classical Erdős–Borwein constant’s irrationality is a one-line corollary.

This formalises the positive-integer-base scope of Erdős’s classical 1948 full-support theorem, which already covered integer bases of absolute value greater than one. The base-two Erdős–Borwein constant is a particular case; the Lean theorem is recorded with no open hypotheses.

Structured infinite supports

Beyond full support ( $A = {1, 2, 3, \dots}$ , i.e. $m = 1$ below), the following named infinite families are proved irrational for every integer base $b \geq 2$ . The purely periodic, eventually periodic, residue-class and odd supports follow from the periodic theorem of Luca and Tachiya [3]; these cases also have Lean proofs (margin):

Theorem 5.5 (Purely periodic support). For integers $b \geq 2$ and $m \geq 1$ , and every $m$ -periodic $A \subseteq N$ (i.e. $n + m \in A \Leftrightarrow n \in A$ for all $n$ ) containing a positive element, the support series $\sum_{a \in A, a \geq 1} (b^{a} - 1)^{- 1}$ is irrational. Hypotheses: $b \geq 2$ , $m \geq 1$ , $A$ $m$ -periodic, $\exists a > 0, a \in A$ .

Theorem 5.6 (Eventually periodic support). For integers $b \geq 2$ and $m \geq 1$ , if $A \subseteq N$ is infinite and its membership is $m$ -periodic from some threshold $N_{0}$ on (i.e. $n + m \in A \Leftrightarrow n \in A$ for all $n \geq N_{0}$ ), then $\sum_{a \in A, a \geq 1} (b^{a} - 1)^{- 1}$ is irrational, by transferring irrationality across the finite symmetric difference from the shifted purely periodic set $A_{pure} := {n : n + m N_{0} \in A}$ . Hypotheses: $b \geq 2$ , $m \geq 1$ , $N_{0} \in N$ , eventual $m$ -periodicity from $N_{0}$ , $A$ infinite.

Theorem 5.7 (Residue-class support). For integers $b \geq 2$ , $m \geq 1$ and $c$ , the series $\sum_{n \geq 1, n \equiv c (\mod m)} (b^{n} - 1)^{- 1}$ is irrational.

Theorem 5.8 (Odd support). For every integer $b \geq 2$ , the series $\sum_{n \geq 1, n odd} (b^{n} - 1)^{- 1}$ is irrational. This density- $1 / 2$ support is the case treated explicitly by Luca and Tachiya [3].

A support representing $1 / 2$ cannot be eventually periodic, by these irrationality theorems. This is a restriction on a hypothetical representation, not a proof that the actual greedy support is nonperiodic: if the greedy construction fails, it eventually selects every exponent. The periodic-support results therefore decide neither alternative for $1 / 2 \in A$ .

Topology and measure of the achievement set

Theorem 5.9 (Achievement-set topology and measure). $A$ (Definition 5.2) is compact, closed, perfect, totally disconnected, and nowhere dense; its Lebesgue measure is exactly $1$ : $volume (A) = 1$ . Hypotheses: none. Conclusion: $A$ is a Cantor set of positive measure, often called a fat Cantor set. Its measure is $1$ , not the length of its ambient interval $[0, E]$ , where $E \approx 1.6067$ . Strict separation and summability give the compactness, unique coding, and Cantor topology, and Hornich’s strict-tail theorem, as proved by Nitecki [4], gives the measure as $lim_{N} 2^{N} R_{N}$ with $R_{N} = \sum_{n > N} x_{n}$ : each level- $N$ cylinder has length $R_{N}$ , and the $2^{N}$ disjoint cylinders have total length $2^{N} R_{N}$ . The Mersenne-specific input is $2^{N} R_{N} \to 1$ . Other weight sequences require their own tail asymptotic.
Use: Theorem 5.10 below (compactness is exactly what powers every “limit of achieved points is achieved” argument used downstream, including the seam-limit route of Part 2).

Theorem 5.10 (Membership equals greedy survival at every level). For a real $x$ , let $r_{n} (x)$ be its greedy remainder after rank $n$ , and let $R_{n} = \sum_{j > n} w_{j}$ , as in the initial notation. Then

x \in A ⟺ x \geq 0 and r_{n} (x) \leq R_{n} for every n \geq 0.

The inequality says that the unselected tail has enough total mass at every rank. A failure at any one rank excludes membership; the inequalities at all ranks give a representation.

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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