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.

Start here, by the editorial route

In this paper

Sparse and dense supports under the certificate criterion

The statement needed

\begin{matrix} A \subseteq N_{> 0}, A infinite ⟹ \\ c_{A} has arbitrarily late super-logarithmic zero windows \\ or the specified block certificates exist at every precision. \end{matrix}

For a fixed nonempty $A$ , let $a_{0} = min A$ . Every interval of $a_{0}$ consecutive integers contains a multiple of $a_{0}$ , so $c_{A}$ has no zero window longer than $a_{0} - 1$ . The first alternative is therefore impossible, independently of rationality. The cited logarithmic result supportCoeffZeroWindow_length_le_eps_logb does not create a sparse subclass here. The separate lcm-gap theorem irrational_erdosSum_of_lcm_gap remains a valid sufficient criterion, but does not prove this proposed alternative. The proposal would consequently require the following certificates for every infinite support:

\exists N, K, L, C, K \leq L, \forall r \in [1, K] : 2^{r} ∣ c_{A} (N + r),

\sum_{r = K + 1}^{L} c_{A} (N + r) 2^{L - r} \leq C, q (C + N + L + 2) < 2^{L} .

What kind of object would furnish it

The identity $c_{A} = 1_{A} * 1$ is a Dirichlet convolution, and Möbius inversion gives $1_{A} = μ * c_{A}$ . These identities hold for every support, but do not supply the displayed divisibility or tail estimates. Shared factors restrict which residue prescriptions can be realised; they do not make the Chinese remainder theorem inapplicable. For positive moduli $m_{i}$ , a finite system $N \equiv a_{i} (\mod m_{i})$ is solvable precisely when

a_{i} \equiv a_{j} (\mod gcd (m_{i}, m_{j})) for every i, j .

Necessity follows by subtraction. For sufficiency, keep the strongest prime-power modulus for each prime; compatibility makes all weaker prescriptions agree with it, and the ordinary CRT combines the resulting coprime prime powers [23]. For example, $N \equiv 1 (\mod 4)$ and $N \equiv 1 (\mod 6)$ give $N \equiv 1 (\mod 12)$ , whereas $4 ∣ N$ and $2 ∤ N$ are incompatible. A block-certificate construction must satisfy these compatibility conditions and its separate tail budget. Neither Möbius inversion nor CRT alone provides both inputs.

The specific obstacle, proved

The dense branch as stated is false: an explicit, natural, divisor-dense support refutes it. This also corrects an earlier claim that universal #257 reduces with no loss to the existence of certificates for every infinite support.

Proposition 11.4 (Squarefree support: a limit of the certificate method, not an open value). Let $A = {n \geq 2 : n is squarefree}$ . Then

c_{A} (n) = 2^{ω (n)} - 1,

which is odd for every $n \geq 2$ . Consequently neither the digitwise nor the carry-aware divisibility-first block-certificate schema has an instance at any even base.

This does not leave the squarefree value open. Duverney–Tachiya’s Corollary 1.2 and Example 1.1 prove that, for every $h \geq 1$ ,

1, \sum_{n s q u a r e f r e e} \frac{1}{2^{n} - 1}, \dots, \sum_{n s q u a r e f r e e} \frac{1}{2^{h n} - 1}

are linearly independent over $Q$ . After deleting the rational $n = 1$ term, squarefree support is therefore irrational at every base $2^{j}$ . Among the integer bases, the cited theorem leaves the non-powers of two uncovered. This is cited mathematics, not formalised here. Proposition 11.4 is a failure of two proof schemas on a value already known to be irrational.

The dependence on the choice of support is exact. Adjoin $1$ to the support. The series changes by the rational number $(b - 1)^{- 1}$ , so its irrationality is unchanged, while the incidence becomes $2^{ω (n)}$ : ErdosProblems.Erdos257.supportCoeff_squarefreeShiftedSupport and ErdosProblems.Erdos257.irrational_erdosSupportSeries_squarefreeSupport_iff_shifted. At base $2$ , CRT then supplies arbitrarily long opening blocks with $ω (N + r) \geq r$ : ErdosProblems.Erdos257.exists_omega_ge_block and ErdosProblems.Erdos257.exists_digitwise_block_squarefreeShiftedSupport. The middle-window and height conditions remain; no certificate or new irrationality proof follows. The durable conclusion is methodological: test a no-go statement under rational finite changes before attributing it to the underlying value.

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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