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

Reset bounds and runs of right transitions

Definition 12.7 (Reset $\sqrt{-escape}$ from row 10).

SQRTESC := \forall r \geq 10 that are resets, | rem (r + 1) - 2^{r + 1} | > 2^{(r + 5) / 2} (equivalently Δ_{r + 1}^{2} > 2^{r + 5}) .

The conditional implication uses the integer bound $B (r) = 2^{⌊ (r + 4) / 2 ⌋} + 2 r + 3$ described in Theorem 9.46: $B (r)^{2} \leq 2^{r + 5}$ for $r \geq 10$ , while a right branch at the critical crossing would force $| Δ_{r + 1} | \leq B (r)$ . Thus the actual implication chain is

SQRTESC ⟹ the hypothesis of Theorem 6.6 ⟹ HALF .

The last implication is half_mem_mersenneAchievementSet_of_largestSkipLateStepSocket; the first is proved in Theorem 9.46. These are sufficient implications, not an equivalence with half-membership.

For comparison one may ask for

RUNBOUND : \forall reset r \geq 31, L_{r} < (r - 3) / 2,

where $L_{r}$ counts the consecutive right branches after reset $r$ . Along $L$ such steps the exact relation is

w_{r + 1 + L} = 4^{L} w_{r + 1} - \sum_{j = 0}^{L - 1} 4^{L - 1 - j} (4 + p_{r + 1 + j}) .

The pulse sum and the threshold ranges cannot be dropped to claim $RUNBOUND \Leftrightarrow SQRTESC$ . The pointwise run/deviation equivalence stated in earlier source notes is not established by the bound used in the conditional proof. The observation at $r = 7$ (deviation $9$ , square-root threshold $64$ ) is outside both proposed late ranges; that fact alone does not justify replacing the initial threshold $10$ by $31$ .

Finite evidence. Exact integer arithmetic to row $200,000$ shows that the target inequality holds at every reset row $6 \leq r \leq 200,000$ except $r = 7$ ; the maximum observed R-run is $19$ , after the reset at row $158,096$ , against a required threshold there of $\approx 79,046$ , a margin of about $4000$ times. This is a finite verified list; it supplies no witnesses at unbounded depths.

Observation 12.8 (The sign of the reset deviation). At all $1209$ observed resets, the sign of the deviation is perfectly predicted by branch type (M $\Rightarrow$ dev $> 0$ , U $\Rightarrow$ dev $< 0$ , zero exceptions), and the margin over the threshold grows from $1.11$ bits at $r = 14$ to $\sim 1246$ bits by row $2500$ (exact computation). Proving the sign law reduces to the middle inequality $4 rem (r) > p_{r}$ . For an upper reset the sign already follows from $w_{r + 1} = - (4 o_{r} + p_{r}^{+}) < 0$ , with positive overshoot and nonnegative pulse. The middle inequality remains unproved. Even after both signs are known, a separate lower bound for their magnitudes is required for $SQRTESC$ .

Observation 12.9 (Rigidity of dangerous resets). If reset $r$ is dangerous ( $| rem (r + 1) - 2^{r + 1} | \leq 2^{(r + 5) / 2}$ ) and the preceding reset $r_{0}$ has a pure R-run of length $L^{'} = r - r_{0} - 1 > (r + 5) / 4$ , then $w_{r_{0} + 1}$ is confined to at most two adjacent integers, determined by the divisor-pulse stream alone. A chain of $n$ consecutive dangerous resets with long runs forces $n - 1$ nested exact congruence towers. This shows the failure mode is a tower of exact integer coincidences, not drift; it is an ordinary argument, and it does not by itself exclude the two pinned values; an isolated dangerous reset after a short run is untouched by this theorem.

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.