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

Three middle-cell exclusions and the right-branch bound

Definition 12.10. The two-sided cell condition excludes three integer values of the middle coordinate, $4 \cdot rem (s) - belowPulse (s) - 4 \notin {- 3, - 2, - 1}$ , plus one right-pulse-leak bound $4 \cdot overshoot (s) + abovePulse (s) \leq 2^{s + 2}$ under $overshoot (s) \leq 2^{s}$ (SeamTwoSidedDyadicCellEscape).

Granted this, induction from a base case at row $5$ , checked by direct computation, propagates $\forall s \geq 5, rem (s) \leq 2^{s} \lor overshoot (s) \leq 2^{s}$ (SeamTwoSidedDyadicCellEscape.twoSided, given the hypothesis). The final-middle theorem excludes $- 3$ at a middle row $D \geq 13$ followed by an all-right tail (finalMiddleCell_neg_three_not_last). It does not supply the $- 3$ exclusion on each eligible non-carry middle branch required by this induction. To complete this route, the three exclusions must hold on those branches of the actual orbit, without an all-right-tail assumption, and the right-branch bound must hold under its own antecedents. Explicitly, at every row $s \geq 5$ where the upper carry does not occur, that is, $4 o_{s} + p_{s}^{+} > 2^{s + 1}$ , the required implications are

\begin{matrix} 4 rem (s) + 2^{s + 1} - p_{s}^{-} < 2^{s + 2} + 4 ⟹ C_{s} \notin {- 3, - 2, - 1}; \\ 4 rem (s) + 2^{s + 1} - p_{s}^{-} \geq 2^{s + 2} + 4, o_{s} \leq 2^{s} ⟹ 4 o_{s} + p_{s}^{+} \leq 2^{s + 2} . \end{matrix}

Here $C_{s} = 4 rem (s) - p_{s}^{-} - 4$ , and $o_{s}, p_{s}^{-}, p_{s}^{+}$ are the overshoot and correction terms of Theorem 5.16. These conditions exclude specified integer values in one coordinate. The square-root reset condition instead bounds the magnitude of a different quantity by a rank-dependent threshold. Their numerical widths do not order the two hypotheses, and no implication between them is established here.

A distinct sufficient condition concerns upper resets only. For every $d \geq 13$ at which the upper carry occurs, set $E_{d} = 4 overshoot (d) + abovePulse (d)$ and require, for every $0 \leq j \leq d$ ,

2^{d - j + 1} < E_{d} or E_{d} + 2 (d + j) \leq 2^{d - j + 1} .

This is the upper-reset band condition of Theorem 6.10 (SeamUpperResetDyadicBandEscape). It is unproved in general, and verified by exact finite computation for rows $13 \leq d \leq 30$ (seamUpperResetDyadicBandEscape_through_thirty). This band has linear width $O (d)$ , whereas the other bound uses a width of order $2^{d / 2}$ . The quantities being bounded are different, so this comparison of widths does not prove a logical implication or strict weakening. The linear-width condition suffices for half-membership (half_mem_mersenneAchievementSet_of_upperResetDyadicBandEscape). It is proved logically equivalent to a single-critical-index check per row (seamUpperResetCriticalBandEscape_iff), which is a genuine quantifier-complexity reduction, not a strength reduction.

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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