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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Finite ranges and unbounded parameters

Half-membership follows from exact rows at unbounded depths or from infinitely many greedy skips. The reset criterion has a different form: it assumes a bound at every reset beyond a threshold. We compare these quantifiers before using the shared transition identities. A finite witness cannot replace either unbounded existence or a uniform bound.

For example, $\forall n \geq N_{0}, P (n)$ yields cofinal witnesses to $P$ , by taking $n = max {N, N_{0}}$ . An implication $H (n) \Rightarrow C (n)$ yields no such witness before $H (n)$ is proved, and a finite verified band says nothing about later indices. Compatibility between cofinal witnesses is a further condition only where it is imposed explicitly.

Table tab:scale-ladder records both the exact parameter ranges and the reasons for the assertions: ordinary or formal proofs, cited theorems, finite calculations and unproved hypotheses. The lower rows include proved conditional implications as well as open antecedents. Neither their truth nor their evidence is determined by the quantifier prefix alone.

Quantifiers for the stated results. The groups separate finite tests, uniform implications, unbounded witness conditions and all-reset hypotheses. A theorem conditional on an unproved hypothesis is not a proof of that hypothesis.
Result	Quantifier prefix
Result	Quantifier prefix

Seed for continuation from $n \geq 6$ : support ${2, 3, 6}$ satisfies $Q = 2^{5} - 1$ (exact computation). `exactLocalMersenneHalfRow_six`	$n = 6$
One-point countermodel: the double-or-recycle transition shape alone, even seeded, does not give exact rows at unbounded depths. `boundedDoubleOrRecycleModel_not_cofinal`	$n = 6$ (free-standing model)
The combined-residue-mass $\leq 1$ sufficient test for sharp capacity is not necessary: example $D = {2, 3}$ , $c = 5$ (exact computation). `splitFractionMass_one_bound_not_necessary_fixture`	$D = {2, 3}, c = 5$
Exact residual: $\frac{1}{2} - W ({2, 3, 6, 7}) = \frac{1}{16002}$ (exact computation). `half_sub_four_term_prefix_eq`	none (numeric instance)
Closed-form Möbius–Lambert value $\sum_{d \geq 1} μ (d) / (2^{d} - 1) = 1 / 2$ (classical identity, cited). `tsum_moebius_div_two_pow_sub_one_eq_half`	none (closed-form value)
The one natural negative-Möbius candidate support overshoots: $1 / 2 < \sum_{d \in N} 1 / (2^{d} - 1)$ for $N = {d \geq 2 : μ (d) = - 1}$ . `half_lt_tsum_negativeMobius`	one fixed candidate support $N$

From an exact row, either repeated doubling is available or an exact row is obtained at some $2 c - 2$ with $c \leq n$ ; these alternatives need not be disjoint. `exactLocalMersenneHalfRow_double_or_recycle`	$\forall n \geq 6$ , given a row at $n$ : $(\dots) \lor \exists c \leq n (\dots)$
Every above-half bounded support recycles into a genuine exact row at some $2 c - 2$ with $c \leq n$ . `exists_skippedCoreExactRow_of_value_above`	$E \subseteq [2, n]$ , $X_{E} (2) > 1 / 2$ : $\exists c \leq n$
(A cross-check for Problem #249.) $\sum φ (n) / 2^{n}$ matches no rational whose denominator divides $2^{14} (2^{h} - 1)$ , for some $1 \leq h \leq 16$ (exact computation). `totient_series_ne_rat_of_den_dvd_upto_sixteen`	$1 \leq h \leq 16$ (explicit bound)
Truncation-rung ladder: $HalfRung (J)$ proved for every $3 \leq J \leq 22$ via the finite decision procedure (§8, Theorem 8.9); ordinary argument with exact computation (working note, program).	$3 \leq J \leq 22$ (finite table, exhaustive)

Membership equals greedy survival. `mem_mersenneAchievementSet_iff_greedy_survival`	$\forall x \geq 0$ , membership iff survival at every rank
Membership iff every actually skipped rank survives. `no_lastHalfGreedySkip_iff_every_skip_survives`	$\forall M \in$ skipped support
Every crossing core is forced to be the canonical rational half-greedy prefix. `eq_halfGreedyPrefixSupport_of_critical_crossing`	$c \geq 4$ , $D \subseteq [2, c)$ , $0 < 1 / 2 - X_{D} (2) < w_{c}$
Largest-skip conditional implication used in the reset argument. `half_mem_mersenneAchievementSet_of_largestSkipLateStepSocket`	$\forall s \geq 14$ , given the hypothesis
Band avoidance verified by exact computation at reset indices $13 \leq r \leq 30$ , using successor remainders at indices $14, \dots, 31$ . This is a finite range of the conditional hypothesis. `seamUpperResetDyadicBandEscape_through_thirty`	$13 \leq r \leq 30$ , exhaustive
Critical-band-index reduction: a $(d + 1)$ -way check collapses to one nearest-boundary check. `half_mem_mersenneAchievementSet_of_upperResetCriticalBandEscape`	$\forall d, E$ with $E \leq 2^{d + 1}$
Parity obstruction: no finite support has value $1 / 2$ . `finite_boolSupport_ne_half`	$\forall A$ finite, $0 \notin A$
Full-support irrationality at every integer base. `irrational_erdosSum_full_support`; base two: `irrational_erdosBorwein_series`	$\forall b \geq 2$
Erdős 1968: any infinite, pairwise-coprime, reciprocal-summable index set gives an irrational support series. `irrational_erdosSupportSeries_pairwise_coprime`	$\forall b \geq 2 \forall A$ (infinite, coprime, summable)
Block-certificate criterion, independent of the problem: a weighted-coefficient series is irrational given a divisibility and window certificate at every precision. `irrational_coeff_series_of_weighted_coeff_block_certificates`	$\forall b \geq 2 \forall c (\forall m, c (m) \leq m)$ , given $\forall q \exists N, K, L, C$
Finite greedy-prefix denominators are odd (source of the parity boundary exclusion, §8). `halfGreedyPrefixDenominator_odd`	$\forall n$


The exact-row existence condition: exact rows recur at arbitrarily large endpoints, with no coherence required between witnesses (unproved). `CofinalExactLocalMersenneHalfRows`	$\forall N \exists n \geq N$ : an exact row at depth $n$
The preceding condition implies $1 / 2 \in A$ by compactness (theorem proved; hypothesis open). `half_mem_mersenneAchievementSet_of_cofinalExactLocalRows`	given $\forall N \exists n \geq N (\dots)$
An equivalent membership condition: the canonical rational half-greedy orbit itself has infinitely many positive skips (unproved). `CofinalPositiveHalfGreedySkips`	$\forall N \exists c \geq max (N, 4), 0 < rem (c - 1) < w_{c}$
Infinitely many positive greedy skips imply $1 / 2 \in A$ (theorem proved; hypothesis open). `half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips`	given $\forall N \exists c \geq N (\dots)$
Membership iff the greedy skip set is infinite (equivalence proved; both sides assert unboundedly many indices). `half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite`	the skipped support is infinite
Seven equivalent forms of membership (Theorem 6.32): the integer-greedy sequence is not eventually always on branch $R$ , the terminal rank is omitted at unboundedly many rows, there are cofinally many skipped ranks, etc. `half_mem_mersenneAchievementSet_iff_not_seamGreedyEventuallyRight`	every right-hand side asserts unboundedly many indices
Ladder dichotomy: either infinitely many rungs survive (#257 false, explicit limit support) or an eventual bad-rank obstruction closes only this route (ordinary argument, Corollary 8 of the truncation-rung note).	open branch (a): $\exists^{\infty} J, HalfRung (J)$
Reset square-root escape, a sufficient all-reset hypothesis (unproved). Theorem 9.46 gives the conditional implication; the run-length comparison is discussed separately in Section 12.5.	$\forall r \geq 10$ that are resets, $\| rem (r + 1) - 2^{r + 1} \| > 2^{(r + 5) / 2}$

Remark 7.1 (Finite checks and quantified conclusions). An earlier review examined eighteen possible implications from recorded results to the required unbounded-witness conditions. It found no such implication. This is a report about those attempts, not a theorem excluding other arguments.

For example, seamUpperResetDyadicBandEscape_through_thirty checks reset indices $13$ through $30$ ; it has no conclusion beyond that range. Likewise, a bound depending on the depth need not produce a witness at a larger depth. Conversely, a proved eventual assertion of the same witness predicate immediately gives cofinal witnesses. What decides the matter is the precise premise and conclusion, whatever tactic a finite proof uses.

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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