Reciprocal Mersenne Subseries

Will Cook

Long paper

Reciprocal Mersenne Subseries

Erdős #257 By Will Cook 30 September 2026 Authorship, AI use and citation

PDFOpen the PDF, 152 pages TeXLaTeX source Source on GitHub Problem page

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.

Start here, by the editorial route

In this paper

The scalar-localisation height obstruction

Lemma 10.13 (Denominator complement survives scaling). For $x : Q$ , $c : Z$ , $H : N$ : if $H ∣ x . den$ and $(c \cdot x) . den ∣ H$ ; i.e. multiplying by the integer $c$ shrinks the displayed denominator down into $H$ ; then the complementary denominator factor $x . den / H$ divides $c$ :

H ∣ x . den \land (c \cdot x) . den ∣ H ⟹ x . den / H ∣ | c | .

Equivalently, writing $D = x . den / H$ , we have $D ∣ c$ , so $t = c / D$ is an integer and $H c x = t x . num$ . Here $H > 0$ follows from $H ∣ x . den$ , since the reduced denominator is positive.

Scope. This is elementary rational arithmetic, independent of the problem. If $c \neq 0$ , divisibility gives the size bound $| c | \geq x . den / H$ . The nonzero condition is essential: $c = 0$ clears every denominator and satisfies the divisibility conclusion without any positive lower bound on $| c |$ . The lemma therefore constrains denominator clearing by a nonzero bounded multiplier, not arbitrary multiplication. Object or representation: about a representation ; pure denominator bookkeeping for $Q$ , with zero problem-specific content; the upstream primitive is the domain-neutral extraction this lemma builds on.

Corollary 10.14 (Mersenne specialisation). Let $x$ be a positive rational number and let $r \geq 0$ , $n \geq 1$ be integers. If $2^{r} ∣ x . num . natAbs$ and $x < 2 / (2^{n} - 1)$ , then $2^{r} \cdot (2^{n} - 1) < 2 \cdot x . den$ ; a numerator $2$ -power lower bound plus a Mersenne-scale upper bound on $x$ together force a denominator lower bound, without introducing a global prefix LCM. To see the inequality directly, write $x = a / b$ in lowest terms with $a, b > 0$ . The hypotheses give $a \geq 2^{r}$ and $a (2^{n} - 1) < 2 b$ . The conclusion follows by substitution. An application of Lemma 10.13 must in addition supply its denominator-divisibility assumptions.

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.

Prefer the manuscript?

PDFOpen the PDF, 152 pages TeXOpen the LaTeX source Problem page

Equations typeset from the exact TeX. Rendered from the LaTeX manuscript source with file fingerprint sha256:47cf6a152738f17c. Its file fingerprint matched the published source record, so the PDF above, the LaTeX, and this page are one manuscript. Redeploying the site regenerates this page from whatever the public repository holds at that moment.