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

Uniqueness of the specified greedy words

Proposition 2.2 (Uniqueness under the stated finite conditions). Put $G = greedyMersenneSupport (1 / 2)$ , $w_{n} = (2^{n} - 1)^{- 1}$ and $R_{d} = \sum_{n > d} w_{n}$ . The following are three separate uniqueness statements.

If $D \subseteq {1, \dots, d}$ and $X_{D} (2) \leq 1 / 2 \leq X_{D} (2) + R_{d}$ , then $D = G \cap {1, \dots, d}$ .
If $c \geq 4$ , $D \subseteq {2, \dots, c - 1}$ and $0 < 1 / 2 - X_{D} (2) < w_{c}$ , then $D = G \cap {1, \dots, c - 1}$ .
Let positive integer weights $v_{1}, \dots, v_{k}$ satisfy $v_{i} \geq g + \sum_{j > i} v_{j}$ for a positive integer $g$ . For a nonnegative integer target $T$ , let $y^{*}$ be the word obtained by visiting these weights in order and taking each when it fits. An admissible Boolean word $y$ has $0 \leq T - \sum_{i} y_{i} v_{i} < g$ if and only if $y = y^{*}$ and $0 \leq T - \sum_{i} y_{i}^{*} v_{i} < g$ .

Part (iii) gives uniqueness under the small-remainder condition, not existence. Theorem 6.100 gives a one-weight counterexample to dropping that condition.

Proof. For (i), truncate $D$ at each successive rank. The target remains in the corresponding prefix interval because all discarded terms are bounded by the complete tail. The two child intervals are disjoint: the lower one forces a greedy skip and the upper one forces a greedy take. Induction identifies every digit. Part (ii) implies (i) at depth $c - 1$ , since $w_{c} < R_{c - 1}$ . For (iii), consider the first disagreement with the integer greedy word. Taking a rejected weight would exceed $T$ ; rejecting an accepted weight leaves a deficit at least $g$ , even after taking every later weight. The converse is immediate. ◻

Excludes at fixed depth. Under the three canonicalisation hypotheses just stated, a proof cannot gain freedom by choosing a different admissible word for the same target and depth. This says nothing about families satisfying different approximation conditions, or about averaging over depths or auxiliary data.

What the finite tests show. Three different calculations are recorded here. For unrestricted prefix residues, the proposed $2^{t / 2}$ exclusion radius is reported to fail in $74 %$ of the tested ranks $t \in [8, 26]$ . Here the finite test uses the truncated, scaled fractional parts

b_{d} (t) = ⌊ 2^{t} {\frac{2^{t}}{2^{d} - 1}} ⌋, {y} = y - ⌊ y ⌋ .

At $t = 15$ , the values for $d = 2, 3, 5, 6, 10$ are respectively $21845, 4681, 1057, 4161, 1025$ . Their sum is $32769 = 2^{15} + 1$ , so the sum has circular distance one from $0$ modulo $2^{15}$ . This is a statement about these integer truncations, not the exact fractional-part sum. A heuristic count of near-hits has size $2^{t / 6}$ ; that estimate is not a theorem about the greedy support.

Long prescribed digit strings also occur for unrestricted finite words: the recorded examples at $t = 20$ and $t = 22$ have respectively eighteen leading ones and twenty leading zeros. A separate construction supplies about $t / 12$ independently adjustable bits, with the reported finite check extending to $t = 601$ . These examples disprove the proposed static exclusions, not an assertion about the actual greedy word.

The integer-window search reported no counterexamples for $t \leq 22$ or at $t = 40, 42, 45$ over the tested cut positions. That absence is a finite observation. Proposition 2.2 explains why certain specified approximation conditions leave only the greedy word; it does not identify all three searches with one another or rule out averaging over other data.

Remark 2.3 (A sufficient condition that is not necessary). The natural sufficient condition via fractional split-mass is not necessary on real crossing cores: the example $D = {2, 3}$ , $c = 5$ is an explicit counterexample (splitFractionMass_one_bound_not_necessary_fixture). The example refutes necessity of that sufficient condition. It does not refute using the condition on a class of cores for which it can be proved.

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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