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 the obstructions do and do not show

We begin with uniqueness at a fixed target and depth, then compare the quantifiers required by the different estimates. The counterexamples identify a failed hypothesis or a missing uniformity, while the equivalences retain the full difficulty of their target statements.

Observation 2.1 (Uniqueness within the fixed-target coding). Each Mersenne weight exceeds the sum of all subsequent weights. Consequently the digit coding is injective (Theorem 1.8): a represented target has only one support, recovered by the greedy rule. For the integer weights, both the scale and the cutoff matter. If $2 \leq k \leq M$ and $q_{M, d} = ⌊ 2^{M} / (2^{d} - 1) ⌋$ for $2 \leq d \leq k$ , then

q_{M, d} - \sum_{e = d + 1}^{k} q_{M, e} \geq 2^{M - k} .

Indeed, with $w_{d} = (2^{d} - 1)^{- 1}$ and $R_{k} = \sum_{e > k} w_{e}$ ,

w_{d} - \sum_{e = d + 1}^{k} w_{e} = w_{d} - R_{d} + R_{k} > R_{k} > 2^{- k} .

Rounding the head down loses less than one, while rounding the later terms down can only increase the difference. The resulting integer is greater than $2^{M - k} - 1$ , proving the bound. Comparing two words at their first differing rank gives the same separation for their sums. The rows with $M = 2 s$ , $k = s - 1$ and $s \geq 3$ therefore have gap at least $2^{s + 1}$ ; the full rows with $k = M$ have gap at least $1$ . The first bound must not be applied after enlarging the cutoff: at $M = 6$ , adjoining rank $3$ introduces the subset sum $q_{6, 3} = 9 < 16 = 2^{3 + 1}$ .

When a finite construction is proved to encode a fixed greedy target, its choices are constrained by that support. Injectivity alone does not identify an arbitrary finite approximation with a real greedy prefix. Nor does it exclude averaging over indices, moduli, nearby targets or auxiliary constructions.
Sources: weightedBoolSum_separated, three_mul_tailWeight_add_exactLateGap_eq_three_mul_headWeight, positiveMersenneDigitValue_injective.

The six limitations below require separate arguments. Fixed-target uniqueness explains canonicalisation (2.2); the recorded identities determine which sufficient conditions are equivalent to their goal (2.3). Measure and category (2.6), observation scales (2.4), finite versus infinite membership (2.5), and the hypotheses of the support certificate (2.7) do not follow from injectivity alone.

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.

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