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 statement needed

The first two conditions below each have a conditional implication to half-membership. No implication between them is asserted. The sign condition is weaker than a quantitative margin in its conclusion, but its relation to either membership hypothesis must still be proved.

Form (i), the reset-deviation bound. Write $w_{s} = rem (s) - 2^{s}$ . The actual hypothesis is

\forall r \geq 10 that are upper or middle resets : | w_{r + 1} | > 2^{(r + 5) / 2} .

Equivalently, $| λ_{r + 1} | > 2^{(3 - r) / 2}$ for $λ_{r + 1} = w_{r + 1} / 2^{r + 1}$ . This normalisation concerns distance of the deviation from zero, not distance of an unrelated phase from the nearest integer. The run-length estimate $L_{r} < (r - 3) / 2$ for resets $r \geq 31$ is a proposed comparison target. Its exact relation to the displayed bound requires the pulse terms and the initial range $10 \leq r < 31$ ; it is not stated as an equivalence here. A sublinear estimate gives an eventual ceiling, not automatically the required finite initial cases. An exact computation (saved run) gives a maximum run of length $19$ through row $200,000$ , following the reset at row $158,096$ . That observation is finite evidence, not an asymptotic theorem.

Form (ii), band avoidance, already proved to imply membership. For every upper reset $d \geq 13$ and every $j \leq d$ ,

2^{d - j + 1} < E_{d} \lor E_{d} + 2 (d + j) \leq 2^{d - j + 1}, E_{d} := 4 \cdot {overshoot}_{d} + {abovePulse}_{d}

(SeamUpperResetDyadicBandEscape), with the conditional implication to membership half_mem_mersenneAchievementSet_of_upperResetDyadicBandEscape. The $\forall j \leq d$ band check collapses to a single nearest-boundary index per row by dyadicBandEscape_iff_exists_critical, and rows $13 \leq d \leq 30$ are verified in Lean by seamUpperResetDyadicBandEscape_through_thirty.

Form (iii), the sign of the reset deviation.

\forall reset rows r : r is an M -reset \Rightarrow w_{r + 1} > 0, r is a U -reset \Rightarrow w_{r + 1} < 0 .

A finite recomputation (saved run) verifies these signs at all $1209$ resets in rows $6$ – $2500$ , with $605$ middle and $604$ upper resets. The upper sign follows directly from the exact branch identity below. The middle sign remains an unproved assertion about the orbit; neither sign alone gives the required lower bound on the magnitude.

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.