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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The size of the required error bounds

One sufficient half-membership argument requires the reset inequality

| rem (r + 1) - 2^{r + 1} | > 2^{(r + 5) / 2} (r \geq 10 an upper or middle reset) .

For the normalised deviation $λ_{r + 1} = (rem (r + 1) - 2^{r + 1}) / 2^{r + 1}$ , this is $| λ_{r + 1} | > 2^{(3 - r) / 2}$ . A near-integer estimate for a weighted divisor sum is a proposed way to obtain information about this inequality, not an equivalent condition established here. Section 11.1 displays the exact recurrence and the additional, changing skip-set term. Each obstruction below has its own hypotheses; none excludes every method at the stated scale.

Class (a): fixed precision or bounded state. The following three statements concern abstract recurrence models. Their conclusions apply to the Mersenne problem only within the specified models; they do not use its full divisor-count constraints.

Proposition 2.8 (Limits of the specified local summaries). (i) After $L$ common steps the endpoint residue mod $2^{L}$ of an affine binary orbit is independent of the initial carry: $u (L) - v (L) = 2^{L} (u_{0} - v_{0})$ . (ii) Fix $m \geq 2$ and put $h = ⌊ (m + 1) / 2 ⌋$ . Consider the coefficient sequences

c_{r} (m) = h - r, c_{r} (m + 1) = 2 r, c_{r} (j) = 0 (j \neq m, m + 1), 0 \leq r \leq h .

Their binary sums are all $h 2^{- m}$ , and their scaled tails before position $m$ agree, but their scaled tails immediately after $m$ are $r$ . Thus any exact label-and-decoder system recovering those latter tails must have at least $h + 1 = ⌊ (m + 1) / 2 ⌋ + 1$ labels. A state determined only by the common preceding history, with no new input, cannot distinguish them. (iii) For any starting carry and any finite list of prescribed valuations and odd unit residues modulo $2^{u}$ , with $u \geq 1$ , there are integer input coefficients with those data whose successor carries lie in their prescribed centred intervals. The unrestricted higher bits of the input coefficients may be chosen separately at each step, as in Proposition 10.4, so these fixed-precision symbols alone do not exclude a completion in that specified family. This says nothing about extra constraints imposed by divisor counts from one common support.

For an integer recurrence $J (M + 1) = 2 J (M) - δ (M + 1)$ , reduction modulo $2$ gives $J (M + 1) \equiv δ (M + 1) (\mod 2)$ . For the particular recurrence $K (M + 1) = 2 K (M) - (τ (M + 1) - 1)$ , with $K (2) = 1$ , this gives, for $M \geq 2$ , that $K (M)$ is even exactly when $M$ is a square: divisors pair off except at a square. This is a genuine restriction at a specified index, but it is already encoded in the known forcing term. It supplies no additional run-length estimate by itself.

Class (b): global averages and Diophantine data.

Proposition 2.9 (A bound for the reduced denominator). Let $n \geq 2$ , let ${Skip}_{n} \subseteq {2, \dots, n - 1}$ , and let $D_{n}$ be the reduced denominator of the associated finite sum. Since $D_{n}$ divides the product of its Mersenne denominators,

\log_{2} D_{n} \leq \sum_{d \in {Skip}_{n}} \log_{2} (2^{d} - 1) \leq \sum_{d \in {Skip}_{n}} d \leq \frac{n (n - 1)}{2} - 1.

The middle inequality is strict when the skip set is nonempty; for an empty skip set both sums are zero. This is an upper bound, not a quadratic asymptotic for $D_{n}$ . Combined with the cited fixed irrationality-measure exponent, it does not supply the required $2^{- 3 n / 2}$ -scale estimate. Even an asymptotic for the exponent sum would need additional information about the skip distribution.

Scope of these obstructions. The common pre-index history in Proposition 2.8(ii) does not determine the post-index tail in its countermodel family. This does not exclude a state using additional arithmetic input from the actual support. The displayed denominator bound combined with the cited fixed irrationality-measure exponent does not yield the required run bound. These calculations do not exclude averaging over shifts, residues, or observation scales, nor a sharper analysis of the actual skip distribution.

Prime steps and bounded windows. In the universal recurrence, a prime step subtracts $1$ , so it sends $K (M)$ to $2 K (M) - 1$ ; it is not exact doubling. Observation 10.1 gives an explicit frozen-row drift with values between $2$ and $6$ at the eleven indices $607, \dots, 617$ , including the three prime indices $607$ , $613$ , and $617$ . This finite example shows that prime steps can occur within a small-value window. It neither proves statistical independence nor rules out every argument using prime-counting information.

Remark 2.10 (Status of the exhaustion claim). The preceding counterexamples concern the specified finite summaries and estimates. They do not classify all available tools, even within the broader categories of local information and global averages. No exhaustion claim is used in a subsequent deduction.

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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