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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A quotient bound at a crossing

Theorem 6.70 (A sufficient quotient bound at a crossing). Consider the following condition on finite sets $D$ and integers $c \geq 4$ :

\begin{matrix} D \subseteq {2, \dots, c - 1}, X_{D} (2) < \frac{1}{2} < X_{D} (2) + w_{c} \\ ⟹ Q (D \cup {c}, 2 c - 2) \geq 2^{2 c - 3} . \end{matrix}

It requires the quotient inequality at every strict crossing of one half by an added weight. Theorem 6.93 identifies this inequality with the binary bound needed in the finite construction. If the condition holds for all such $D, c$ , the induction below produces exact sums at unbounded depths and hence an infinite support of value $1 / 2$ . The universal crossing condition itself is unproved.

Theorem 6.71 (Induction from the depth-six example). An already-formalised induction from the endpoint-six seed (Definition 6.97) consumes Theorem 6.70 at every step: each protected exact row either doubles below half (unconditional) or recycles at its first crossing rank $e >$ cutoff, giving endpoint $2 e - 2 >$ previous endpoint; protection (endpoint $< 2 \cdot$ cutoff, new ranks $>$ cutoff) is exactly what converts the non-growing recycle endpoint of the bare dichotomy (Theorem 6.127) into strict progress. The below-half branch never fires twice from the seed arithmetic, so the supply is needed at essentially every step. Its conclusion is the cofinal exact-row condition of Definition 6.66.

Theorem 6.72 (The quotient condition on greedy prefixes). Let $D_{c} = G \cap {2, \dots, c - 1}$ . The condition on the actual greedy prefixes is

\begin{matrix} c \geq 4, c \notin G \\ ⟹ Q (D_{c} \cup {c}, 2 c - 2) \geq 2^{2 c - 3} . \end{matrix}

It asks for the quotient bound at every skipped rank of the specified half-greedy orbit. This restricts the crossing test of Theorem 6.70 to its actual greedy prefixes. The finite uniqueness results justify that restriction where the cited induction uses it; they do not prove the displayed inequality.

Theorem 6.73 (Two consecutive skips). If $c \geq 6$ and the real half-greedy rule skips both $c$ and $c + 1$ , put $D = G \cap {2, \dots, c - 1}$ . Then

S (D, 1, 2 c - 3) < 2^{c - 3} .

The one-step quotient recurrence gives $S (D, 1, 2 c - 2) < 2^{c - 2}$ , so Theorem 6.75 supplies an exact row at depth $2 c - 2$ whose new ranks are all greater than $c$ . The hypothesis concerns this pair of skipped ranks; no unbounded sequence of such pairs is proved here. For the stated precritical-suffix condition at every skipped rank, the remaining tests are the skip-then-take cases, with ranks $4$ and $5$ handled separately in the cited proof.

Proof. The two skips imply $0 < ρ = 1 / 2 - X_{D} (2) < w_{c + 1}$ . Put $M = 2 c - 3$ . The fractional-part error in $Q (D, M)$ is less than $c - 2$ , because $D \subseteq {2, \dots, c - 1}$ . Therefore

S (D, 1, M) < \frac{2^{2 c - 3}}{2^{c + 1} - 1} + c - 3 < 2^{c - 4} + c - 2 \leq 2^{c - 3},

where the last inequality holds for $c \geq 6$ . The suffix is nonnegative because $X_{D} (2) < 1 / 2$ , so its next-step identity is $S (D, 1, M + 1) = 2 S (D, 1, M) + 1 - c_{D} (M + 1)$ . Integrality gives the required strict bound at $M + 1$ . ◻

Theorem 6.74 (A later skip after a selected block). Suppose the real half-greedy rule skips rank $c$ , takes $c + 1, \dots, c + t - 1$ , and skips $c + t$ . If

0 < t \leq c - 3, c - 2 \leq 2^{c - t - 3},

then the precritical suffix bound at $c$ holds. It yields sharp capacity and hence an exact row at depth $2 c - 2$ . The case $t = 1$ includes Theorem 6.73; the proof also permits longer selected blocks. The arithmetic condition is precisely $t \leq c - 3 - ⌈ \log_{2} (c - 2) ⌉$ . Its role in the proof is to make the dyadic allowance $2^{c - t - 3}$ cover the bound $| D | \leq c - 2$ for the earlier selected support.

To obtain cofinal exact rows by this result, the stated gap condition must hold at cofinally many skipped ranks $c$ , not just at one rank or throughout a finite sample. A finite empirical skip frequency alone gives no such pointwise gap bound.

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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