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

The integer update in a middle interval

The following identities explain the two-sided bound and the reset-band condition used above. They also distinguish the signed middle coordinate $C_{s} = 4 rem (s) - p_{s}^{-} - 4$ from the nonnegative label in the next estimate.

Proposition 6.140 (The unsafe middle range is exactly three integers). For $s \geq 5$ , write $R = rem (s)$ and $P = p_{s}^{-}$ . Since $4 R - P - 4$ is an integer,

\neg (4 R - P - 4 \leq - 4 \lor 0 \leq 4 R - P - 4) ⟺ 4 R - P - 4 \in {- 3, - 2, - 1}

The complement of the two safe integer ranges is precisely ${- 3, - 2, - 1}$ . The conclusion is only a classification; excluding any of these values requires an additional argument.

For a row $s \geq 5$ , set

E_{s} = \sum_{d \in D_{s}} {\frac{4^{s}}{2^{d} - 1}}, Z_{s} = rem (s) - E_{s}, c = 4 rem (s) - p_{s}^{-} = C_{s} + 4.

Here braces denote fractional part. Assume $c \geq 0$ , as in the linked statements below. In particular, the exceptional signed values $C_{s} = - 3, - 2, - 1$ correspond to $c = 1, 2, 3$ , not to negative labels. The floor-quotient recurrence gives $p_{s}^{-} \leq 4 E_{s}$ , and hence

4 Z_{s} \leq c .

This inequality has a Lean proof (seamMiddleCell_four_mul_floorZ_le). On a middle transition $D_{s + 1} = D_{s}$ . Writing $ρ = 1 / 2 - X_{D_{s}} (2)$ , the exact quotient identity therefore gives

4^{s + 1} ρ = 2^{s + 2} + 4 Z_{s} \leq 2^{s + 2} + c,

also proved in Lean (seamMiddleCell_scaledRationalRemainder_le).

For $C_{s} = - 3$ , the correct label is $c = 1$ . This estimate does have a local consequence: since $4^{s + 1} R_{s} > 2^{s + 2} + 4 / 3$ , it gives $ρ < R_{s}$ . Thus the completed support $D_{s} \cup {s + 1, s + 2, \dots}$ has value greater than $1 / 2$ . It does not prove that every later greedy step survives. For $s \geq 13$ , under the additional all-right-tail hypothesis, the same completed support would have value less than $1 / 2$ , giving the contradiction in finalMiddleCell_neg_three_not_last. That tail hypothesis must not be dropped. For $c = 2, 3$ , the displayed upper bound alone no longer gives $ρ < R_{s}$ .

A parallel equivalence holds for the right branch: an overshoot witness $\leq 2^{s}$ produces a pulse leak above $2^{s + 2}$ exactly when the overshoot lies within an explicit distance $k$ of $2^{s}$ with $4 k < abovePulse$ (seamRightPulseLeak_iff_distanceCell). The remainder left by the integer greedy rule with the row- $s$ quotient weights and target $2^{2 s - 1}$ (seamPureHalfPrefixRemainder) has an exact case-split formula (seamPureHalfPrefixRemainder_eq_ite), and a bound on it is proved equivalent to the two-sided bound (seamPureHalfPrefixRemainder_le_iff_twoSided).

Observation 6.141 (The reset-band hypothesis has a verified base case). The upper-reset dyadic-band escape condition (discussed after Section 12.5) is reduced by three lemmas to a quarter-scale overshoot exclusion: an equivalence (seamUpperReset_band_iff_successorRemainder_avoids) rewrites the reset-charge band as a successor-remainder avoidance; an identity (upperResetCharge_dyadicDistance_add_abovePulse) rescales it by a factor of $4$ ; and a third lemma (seamUpperReset_band_of_overshoot_band) derives the band from a quarter-scale overshoot exclusion plus the pulse bound $abovePulse \leq 2 (d - 2)$ . Row $d = 13$ , the first row of the late regime, is then proved unconditionally to escape every band simultaneously: its successor remainder is exactly $392$ , by exact computation (seamUpperResetDyadicBandEscape_at_thirteen). This is one verified row of a hypothesis required for every $d \geq 13$ ; it shows only that the base case is not itself an obstruction.

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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