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

Exact finite sums at unbounded depths

Definition 12.6. For every integer $N$ , require some $n \geq max {N, 1}$ and $D \subseteq {2, \dots, n}$ with

\sum_{d \in D} ⌊ \frac{2^{n}}{2^{d} - 1} ⌋ = 2^{n - 1} - 1.

(CofinalExactLocalMersenneHalfRows). This condition is unproved. Supports at different depths need not agree. This is the condition in Definition 6.66; no strict weakening of half-membership is asserted.

The implication from this condition to $HALF$ is proved unconditionally via compactness of $A$ (a continuous image of the space of $0 / 1$ sequences under the digit-coding map, hence closed) (half_mem_mersenneAchievementSet_of_cofinalExactLocalRows). The cited theorem already proves that a representing support exists, because that is what membership of $A$ means. It need not choose a support by a specified algorithm from the row witnesses. An ordinary diagonal subsequence argument gives a compatible limiting support: successively fix each coordinate along an infinite subsequence, then use the uniform tail bound to pass to its value. This uses continuity of the digit map, also recorded in continuous_positiveMersenneDigitValue. The diagonal argument preserves $2 \in A$ , since all sufficiently deep exact rows contain exponent two (two_mem_of_exact_localMersenneQuotient). This describes an ordinary existence proof; no additional formal extraction theorem or effective algorithm from purely existential row data is claimed.

Proposition 2.7(c) already identifies this condition as logically equivalent to $1 / 2 \in A$ . Equivalence of a membership condition, constructive extraction of a coherent support, and a rejected sufficient strengthening are not interchangeable descriptions. Absence of a Lean extraction theorem does not make this condition a strictly weaker substitute for $HALF$ . The remaining work here is to supply exact rows at unbounded depths, or any equivalent membership condition.

A capacity condition for exact sums at unbounded depths

The finite construction uses a $(c - 2)$ -bit capacity test. Write

S D k M := 2^{M - k} - Q D M - 1

(localBinarySuffix), with natural-number subtraction understood as truncated at zero. The loose bound below requires a critical crossing: $c \geq 4$ , $D \subseteq {2, \dots, c - 1}$ , and $0 < 1 / 2 - V (D) < w_{c}$ . A merely below-half core is not enough: $D = \emptyset$ gives $S (D, 1, 2 c - 2) = 2^{2 c - 3} - 1$ . For a critical-crossing core the two bounds are

\begin{aligned} (*) & (loose, proved) S D 1 (2 c - 2) < 2^{c - 1} \\ (* *) & (sharp, needed) S D 1 (2 c - 2) < 2^{c - 2} \end{aligned}

The implication $(*)$ from the critical-crossing hypotheses is proved uniformly in $c$ (localBinarySuffix_two_mul_sub_two_lt_upperHalfCapacity).

$(* *)$ is exactly one bit sharper and, via localBinarySuffix_two_mul_sub_two_lt_criticalCapacity_iff (under the stated core hypotheses), is equivalent to $Q (D \cup {c}, 2 c - 2) \geq 2^{2 c - 3}$ . Here $c$ is the added exponent; the quotient is evaluated at depth $2 c - 2$ , not at depth $c$ . The proof of $(*)$ gives the more informative additive bound $S < 2^{c - 2} + | D |$ before using $| D | \leq c - 2$ . One possible cofinal supply is therefore the following exclusion of a specified integer interval:

\begin{matrix} (capacity band) \forall N \exists c \geq max (N, 4) \exists D \subseteq {2, \dots, c - 1}, \\ 0 < 1 / 2 - V (D) < w_{c}, S (D, 1, 2 c - 2) \notin [2^{c - 2}, 2^{c - 2} + c - 3] . \end{matrix}

The excluded band contains $c - 2$ integers. The sharp capacity bound itself allows $2^{c - 2}$ nonnegative integer values. The numerical widths differ from those in Section 12.5, but the conditions concern different quantities. Width alone does not prove a logical implication between them.

The sharp-fill implication exists_exactRowStrictUpperFill_of_skippedCoreSharpCapacity assumes $(* *)$ for a below-half core $D \subseteq {2, \dots, c - 1}$ , but does not require a critical crossing or identify that core with a real greedy prefix. This gives another formulation in which to seek the required finite rows. It does not establish a strict logical weakening of half-membership, nor does it supply cores satisfying the bound at unbounded ranks. Moreover, the integer gap separation fixes the resulting exact row whenever it exists; the formulation does not create multiple exact rows at a fixed depth.

The complete induction from the endpoint-six seed to exact sums at unbounded depths is already formalised (cofinalExactLocalMersenneHalfRows_of_criticalQuotientSupply); it uses the sharp capacity input, in its reduced canonical form

c \geq 4, c \notin G ⟹ Q ((G \cap {2, \dots, c - 1}) \cup {c}, 2 c - 2) \geq 2^{2 c - 3} .

(HalfGreedySkippedCriticalQuotientSupply), at essentially every step; this input is unproved. This condition is required along the specified greedy prefixes. That restriction on the witnesses does not by itself prove a strict logical ordering with the cofinal condition above.

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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