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 final-skip band formula, and what it does not show

Definition 10.15 (Upper-reset dyadic band escape). At each upper-reset row $d \geq 13$ , write $E_{d} = 4 overshoot (d) + abovePulse (d)$ for the reset expression in the preceding transition formulas. The condition is

for every 0 \leq j \leq d, E_{d} > 2^{d - j + 1} or E_{d} + 2 (d + j) \leq 2^{d - j + 1} .

Thus $E_{d}$ must avoid the interval $(2^{d - j + 1} - 2 (d + j), 2^{d - j + 1}]$ at every indicated scale. Erdos249257.SeamUpperResetDyadicBandEscape. In words: at every upper-reset row $d \geq 13$ , the reset charge must avoid a linear-width forbidden band immediately below every dyadic threshold $2^{d - j + 1}$ , for every $j \leq d$ simultaneously. Granted this hypothesis, Erdos249257.half_mem_mersenneAchievementSet_of_upperResetDyadicBandEscape gives $1 / 2 \in A$ outright.

Proposition 10.16 (Quantifier collapse: one critical index suffices, not $d + 1$ ). Let $d, E \in N$ and assume $E \leq 2^{d + 1}$ . Then the purely combinatorial statement $DyadicBandEscape (d, E) ⟺ \exists j, CriticalDyadicBandIndex (d, E, j) \land E + 2 (d + j) \leq 2^{d - j + 1}$ collapses the $\forall j \in [0, d]$ band-avoidance condition (formally $d + 1$ separate inequalities) to checking exactly one nearest-boundary index $j$ . The range assumption ensures that a dyadic threshold lies at or above $E$ . Choose the smallest such threshold: smaller thresholds are already below $E$ , while larger thresholds have narrower forbidden bands. Without the range assumption all bands escape automatically when $E > 2^{d + 1}$ , but no critical index exists; $(d, E) = (0, 3)$ is the smallest example. Specialised to the concrete seam reset charge, $SeamUpperResetCriticalBandEscape$ is proved logically equivalent to Definition 10.15’s hypothesis. Zero Mersenne/seam content in the core lemma ; pure $(d, E, j)$ arithmetic over powers of 2.

Observation 10.17 (A verified finite range of reset indices). The linked certificate proves the condition in Definition 10.15 for each reset index $13 \leq d \leq 30$ . It uses successor remainders at rows $14$ – $31$ , including $rem (14) = 392$ and $rem (31) = 4,187,487,147$ . Thus the last remainder row is $31$ , but the last reset index covered by this theorem is $30$ . Erdos249257.seamUpperResetDyadicBandEscape_through_thirty.

Remark 10.18 (What this does not show). Definition 10.15 is a $\forall d \geq 13$ statement; Observation 10.17 is a finite verification in the cited Lean source for $13 \leq d \leq 30$ . The finite certificate does not establish the universal hypothesis. Nothing here proves the band condition for all $d \geq 13$ ; it is the unproved hypothesis of Theorem 6.10, stated here in its own form.

The finite theorem supplies no reset index beyond $30$ . It therefore does not prove that the orbit avoids the band at unboundedly many reset indices, let alone at every reset. The universal band condition remains an unproved sufficient hypothesis.

For the branch convention of Theorem 5.16, direct integer recomputation at reset-test indices $6 \leq r \leq 2500$ gives $605$ middle and $604$ upper resets, hence $1209$ resets. There are $1286$ right transitions. The earlier reported count $1264$ does not agree with this convention. The recomputation checks the sign inequalities at these resets, the square-root bound at resets $r \geq 10$ , and the displayed band condition at upper resets $r \geq 13$ , with no failures. This remains a finite ordinary computation, distinct from the cited Lean theorem for $13 \leq r \leq 30$ and from an assertion at every later reset.

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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