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Reciprocal Mersenne Subseries

Erdős #257 By 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(bh(a)−1)) finite has irrational ∑ (ba−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
Nonnegativity of the omitted-tail margin

The next three statements compare the conditions of Theorem 6.13. The first two are equivalent; the third is sufficient for them.

Definition 6.81 (A nonnegativity condition at skipped ranks).

∀ skipped rank n≥3,0≤F(n−1,n).

This condition is not established here. Its usefulness is the following implication: assuming it, a Lean theorem proves 1/2∈A.

HalfGreedySkippedFullShellNonnegative half_mem_mersenneAchievementSet_of_skippedFullShellNonnegative

Theorem 6.82 (An equivalent vanishing condition).

∀ skipped rank n≥3 whose actual word equals the seam-greedy word,the integer remainder at n is zero.

This condition is equivalent to Definition 6.81. At an actual skipped rank, the source proves that F(n−1,n)<0 holds exactly when the real prefix agrees with the integer greedy word and its integer remainder is positive. On agreement, F(n−1,n) is the negative of that nonnegative remainder. Thus nonnegativity of the margin at every skip is equivalent to vanishing of the remainder at every aligned skip. The equivalence is proved; neither condition is established for all required ranks.

Theorem 6.83 (A sufficient lower bound for the integer remainder).

∀ skipped rank n≥3,B(2n)<rem(n),

a sufficient condition for the nonnegativity condition above. Indeed, a negative margin at an actual skip would force 1≤rem(n)≤B(2n), contradicting this bound. This proves the implication, not a converse or strict separation of the conditions.

The displayed lower bound itself is not established.

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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