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 finite certificates decide

Theorem 2.11 (One-sidedness). Non-membership of $1 / 2$ in $A$ has an effectively checkable finite-certificate formulation, hence a $Σ_{1}^{0}$ formulation. Membership has the complementary $Π_{1}^{0}$ formulation. Survival through a tested finite depth alone does not establish membership; a uniform theorem or inductive invariant could. The arithmetical-hierarchy form by itself proves neither undecidability nor the absence of finite proofs.

Proof. The dichotomy $1 / 2 \in A$ or $\exists$ a fatal gap is unconditional (half_mem_mersenneAchievementSet_or_exists_fatal_gap), and the two alternatives are exactly complementary (existsFatalHalfGap_iff_half_not_mem_mersenneAchievementSet), with ExistsFatalHalfGap a finite witness type. In the other direction, certified death implies non-membership (certifiedGreedyMersenneDeath_not_mem, over CertifiedGreedyMersenneDeath; $3 / 4$ is certified dead at level $1$ , lookahead $0$ , by three_fourths_certifiedGreedyMersenneDeath), whereas the absence of a certificate through depth $D$ alone gives no conclusion about later depths. To make the search effective, note that a greedy remainder at a finite rank is rational. The tail $R_{n}$ has rational upper bounds obtained by summing through a cutoff $M$ and bounding the remaining sum by $2^{- M} + (2 / 3) 4^{- M}$ . These upper bounds converge to $R_{n}$ . Thus the strict inequality $r_{n} (1 / 2) > R_{n}$ , when true, is witnessed by some finite cutoff and a comparison of rational numbers. Enumerating ranks and cutoffs gives the asserted finite-certificate search. Membership requires the survival inequality at every rank (Theorem 1.11). Checking a finite initial segment alone does not supply the later inequalities; a separate uniform argument or induction would be needed. ◻

Finite search versus infinite survival. A discovered fatal-gap certificate proves non-membership. Further surviving prefixes give larger verified ranges, but do not establish membership without an argument covering all ranks. This distinction is unchanged whether one increases the geometric truncation depth, the number of quotient rows, or the depth of a grouped computation. The different finite parameters should not be reported as a single verification depth.

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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