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.

Start here, by the editorial route

In this paper

Obstructions and countermodels

The results below test particular proposed implications. A countermodel to an abstract transition rule shows that the rule needs an additional hypothesis; it does not disprove a more structured arithmetic argument. Each entry states the information retained by its model and the conclusion that this information fails to imply.

Theorem 6.127 (A bounded model of the doubling-or-return alternative). The dichotomy $ExactLocalMersenneHalfRow (2 n - 1) \lor \exists c, 4 \leq c \leq n \land ExactLocalMersenneHalfRow (2 c - 2)$ (proved for $n \geq 6$ at Theorem 6.128 below) is not by itself enough for cofinality. Countermodel: $boundedDoubleOrRecycleModel (n) := (n = 6)$ satisfies exactly the same two-branch transition shape (seed at 6, and the model reproduces the $\lor$ shape by always taking the recycle branch with $c = 4$ , conclusion back at $2 \cdot 4 - 2 = 6$ ), yet

\neg (\forall N, \exists n \geq N, boundedDoubleOrRecycleModel (n)) (not cofinal ; only ever true at n = 6) .

The double-or-recycle transition shape, even together with an endpoint-six seed, is logically insufficient to conclude cofinal exact rows. A genuinely new progress input (strict endpoint growth, or an independent cofinality argument) is required. The countermodel retains only the seed and this transition rule. It therefore tests those premises, not additional arithmetic information that a more structured induction may use.

Theorem 6.128 (Doubling or returning to an earlier depth). For $n \geq 6$ , $ExactLocalMersenneHalfRow (n)$ :

ExactLocalMersenneHalfRow (2 n - 1) \lor \exists c, 4 \leq c \leq n \land ExactLocalMersenneHalfRow (2 c - 2) .

Every exact row of depth at least $6$ has at least one of the two asserted continuations. The proof separates the below-half case from the above-half case, but the two existential conclusions need not be exclusive. The returned depth $2 c - 2$ is not claimed to exceed $n$ . The finite Mersenne value cannot equal $1 / 2$ , by Observation 6.138; this says nothing about equality of the returned and original depths. For example, at $n = 6$ , the support ${2, 3, 6}$ is an exact row, and so is ${2, 3, 6, 7, 11}$ at depth $11$ . The second conclusion also holds with $c = 4$ , since $2 c - 2 = 6$ . Thus both conclusions hold in this example, and the second allows no depth increase. What is missing, per Theorem 6.127, is not another dichotomy but strict endpoint progress in the recycle branch, or a proof the below-half branch recurs ; re-deriving this disjunction without an additional progress argument would not establish cofinality.

Proposition 6.129 (A sufficient fractional-mass bound need not hold). Take $D = {2, 3}$ and $c = 5$ . Direct calculation gives

X_{D} (2) = \frac{10}{21} < \frac{1}{2} < \frac{331}{651} = X_{D \cup {5}} (2), S (D, 1, 8) = 6 < 8.

However, the combined fractional mass is

\sum_{d \in D \cup {5}} \frac{2^{8} mod (2^{d} - 1)}{2^{d} - 1} = \frac{757}{651} > 1.

The linked Lean proof checks this example by direct computation. Thus the combined fractional-mass bound by $1$ , which suffices for the sharp capacity estimate, is not necessary for that estimate. Thus the fractional-mass bound cannot hold at every real crossing core. A proof using it would need to handle the exceptional cores separately; the example does not rule out all uses of fractional-mass estimates. Theorem 6.133 identifies the support at a critical crossing with a real greedy prefix, but does not prove the missing bound on those prefixes.

Observation 6.130 (A doubled finite sum can exceed one half). Theorem 6.95’s doubling extension is not preserved iteratively: below-halfness is not preserved under one application of doubling, and demonstrably fails at the very first step. From the identity $value (E) - \frac{1}{2} = (localFractionMass (E, 2 n - 1) - 1) / 2^{2 n - 1}$ , at the seed $D = {2, 3, 6}$ (Definition 6.97) with $M = 11$ the core alone contributes

\frac{2^{11} mod 3}{3} + \frac{2^{11} mod 7}{7} + \frac{2^{11} mod 63}{63} = \frac{2}{3} + \frac{4}{7} + \frac{32}{63} \approx 1.746 > 1,

so the doubled row is strictly above half. Iterating the cheap (doubling) arm is therefore impossible from the only recorded seed; the chain is forced into the capacity-gated recycle arm at step one. What would close the doubling route instead: cofinally many below-half exact rows, $\forall N \exists n \geq N \exists D$ exact at $n$ with $localFractionMass (D, n) < 1$ ; a single such row at each of cofinally many $n$ would make doubling unnecessary; a self-reproducing one would close everything.

exists_exactRowStrictUpperExtension_two_mul_sub_one_of_exact_below

Theorem 6.131 (The returning endpoint need not be larger). Theorem 6.78 is unconditional, but its witness is $\exists c \leq n$ , not $\exists c$ large: $2 c - 2$ may be $\leq n$ , so the endpoint need not grow. The countermodel of Theorem 6.127 is the explicit falsifier of the naive hope that growth comes for free: $boundedDoubleOrRecycleModel (n) := (n = 6)$ satisfies the same transition schema plus a seed and is not cofinal. Growth is recovered only inside $ProtectedExactLocalMersenneRow$ (Theorem 6.71), whose invariants $endpoint < 2 \cdot cutoff$ and $new_above_cutoff$ force $c > cutoff$ hence $2 c - 2 > endpoint$ ; and maintaining those invariants is precisely what needs the strict-upper (sharp capacity) fill of Theorem 6.75 rather than this general recycling theorem.

Observation 6.132 (The skip condition is equivalent to membership). The positivity conjunct of Theorem 6.68’s hypothesis, $0 < r (1 / 2, c - 1)$ , holds unconditionally: the greedy remainder is $1 / 2$ minus a finite Mersenne sum, which is nonzero by the odd-denominator parity fact of Observation 6.138. Hence the hypothesis of Theorem 6.68 holds if and only if the set of ranks skipped by the greedy rule for $1 / 2$ is infinite, which Theorem 6.88 proves equivalent to $1 / 2 \in A$ . Proving Theorem 6.68’s hypothesis would therefore settle the half-target question. This equivalence does not by itself make the hypothesis easier, but it also does not preclude a useful argument in greedy coordinates. Theorem 6.75 offers a different sufficient condition whose finite supports need not be greedy prefixes.

localMersennePrefixValue_halfGreedy_lt_half

Theorem 6.133 (Uniqueness at a critical crossing). For $c \geq 4$ , $D$ bounded $[2, c)$ , below-half, with genuine crossing deficit $\frac{1}{2} - value (D) < w (c)$ :

D = halfGreedyPrefixSupport (c - 1) .

Thus $D = G \cap {1, \dots, c - 1}$ : at a critical crossing the support is fixed by $c$ . The proof uses the strict inequality between each Mersenne weight and the sum of all later weights, as in Theorem 6.100. Consequently, the support quantifier in Theorem 6.70 does not allow arbitrary choices of $D$ . This reduces that hypothesis to a statement about the actual greedy sequence. It does not rule out proving a new bound for that sequence.

Observation 6.134 (Different requirements on finite witnesses). Theorem 6.85 (the cofinal terminal-strip condition) tolerates any support inside the depth- $M$ window but demands the carry inside a $\sim 2 \sqrt{M}$ strip; Theorem 6.75 tolerates a carry up to $2^{c - 2}$ (exponentially looser) but demands the support be confined to ranks $< c$ , because the exact fill encodes the residue in the pure upper window where the Mersenne quotient is exactly $2^{M - d}$ (Proposition 6.77). The two conditions therefore differ in the permitted support as well as the size of the carry. If the square-root witnesses could also be chosen with support in $[2, c)$ at depth $2 c - 3$ , then $2 \sqrt{2 c} + 4 < 2^{c - 2}$ for $c \geq 8$ would give the numerical bound needed for the fill. The required support restriction is not proved here. Comparing the displayed bounds alone does not order the two existence hypotheses.

HalfCarryCofinalTerminalOnlyStrip

Observation 6.135 (An unsafe example in the two-thirds interval). Write a greedy residual as $rem = 1 / R$ ; a skip at rank $k$ is dyadically safe if and only if $R \geq 2^{k}$ . Two unconditional corollaries hold: an integral reciprocal $R$ is never unsafe, and an unsafe run with odd $p, D, q$ forces $p \geq 7$ . Dyadic safety is sufficient but not necessary for the true greedy process to survive: the true test uses the actual tail $\sum_{j > k} w_{j}$ , which exceeds $2^{- k}$ . Nothing here asserts that the actual half-greedy orbit avoids the band: $(p, D, b) = (17, 41, 3)$ is an explicit odd-coprime example that is dyadically unsafe. Theorem 6.120’s criterion $2 u \leq 3 a$ narrows exactly this gap in the single-skip case; whether the fatal region meets a band reachable by the actual greedy trajectory is not settled.

TwoThirdsBand not_twoThirdsBand_of_int seven_le_of_intBand_odd

Theorem 6.136 (Vanishing of the specified linear-channel determinant). Let $V$ be a vector space over $Q$ , let $e : V \to Q$ be linear, and let $(ℓ_{j})_{j \in ι}$ be a finite family of linear functionals vanishing on $\ker e$ . For any vectors $(v_{i})_{i \in ι}$ , the matrix $(ℓ_{j} (v_{i}))_{i, j \in ι}$ has rank at most one, so every square minor of size at least two vanishes. Indeed, the functionals descend to $V / \ker e$ , which has dimension at most one. This elementary linear-algebra argument applies at every matrix size. It does not cover additional functionals that fail to vanish on $\ker e$ .

Observation 6.137 (The additive error in the binary bound). Theorem 6.94’s $2^{c - 1}$ bound and Theorem 6.93’s sharp $2^{c - 2}$ requirement differ by one bit, and the proof of the loose bound shows exactly where: it derives $A < 2^{c - 2} + | D |$ and discards $| D | \leq c - 2 \leq 2^{c - 2}$ . Under this additive upper bound, the remaining assertion is that the integer suffix avoids $[2^{c - 2}, 2^{c - 2} + c - 3]$ . This identifies the unresolved interval; it does not prescribe how to exclude it. A sharper estimate, an arithmetic restriction on the suffix, or another argument could in principle establish the required inequality.

localBinarySuffix_two_mul_sub_two_lt_upperHalfCapacity

Observation 6.138 (Odd denominators of finite sums). For finite $D \subseteq N_{> 0}$ , each denominator $2^{d} - 1$ is odd. The sum $V (D)$ can therefore be written over an odd common denominator; its reduced denominator, being a divisor of that denominator, is odd. In particular, $V (D) \neq 1 / 2$ . This elementary argument supplies the parity fact used in Proposition 6.129 and Proposition 6.99.

finiteErdosSum_den_odd

Theorem 6.139 (Excluding the cell with value minus three). At a middle row $D \geq 13$ followed only by right transitions, $4 rem (D) - p_{D}^{-} - 4 \neq - 3$ , by Theorem 6.44. Its proof uses the nonnegative centred carry for the completed support, not a finite search. The values $- 2, - 1$ remain among the three exceptional negative cells. Excluding them under this extra tail assumption would still not exclude nonnegative values of the coordinate, or establish the all-middle-row and right-branch hypotheses in Theorem 6.9. The complete remaining tail inequality is stated in Remark 10.8.

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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