Problems / erdos-69 / erdos-69--h2-v2--h1
Some rescaled tail times b is not an integer (untyped casts)
erdos-69--h2-v2--h1
superseded
Derived by the compiler.
Superseded by erdos-69--h2-v2--h1-v2. Work continues on the replacement; this statement is kept for its history (D-8). The curator's record: superseded by erdos-69--h2-v2--h1-v2 on revision request targets/erdos-69/nodes/erdos-69--h2-v2--h1/revisions/20260917T182531Z-calib-lead-83dd.yaml (other-with-exhibit; D-8)
$\displaystyle \Bigl(\sum_{n=0}^{\infty}\frac{\uparrow\omega(n)}{2^{n}} = \sum_{p\ \text{prime}}\frac{1}{2^{\uparrow p}-1}\Bigr) \Rightarrow \forall b\in\mathbb{N},\ b>0\ \ \exists N\in\mathbb{N}\ \ \forall z\in\mathbb{Z}:\ \uparrow b\cdot\sum_{j=0}^{\infty}\frac{\uparrow\omega(N+1+j)}{2^{j+1}} \ne \uparrow z$
If the series of ω(n)/2^n equals the sum over primes of 1/(2^p − 1), then for every b ≥ 1 there is an N such that b times the tail ∑_{j≥0} ω(N+1+j)/2^{j+1} is no integer z. Every cast arrow ↑ in the Lean carries no target type, so as written neither the hypothesis nor the tail is stated over ℝ, where the parent's hole states them: their domain is whatever Lean's defaults assign (ℕ for a bare numeral, where division rounds down and 2^p − 1 is truncated subtraction), and ∑' is the tsum (0 when not summable).
Lean
theorem erdos_69__h2_v2__h1 : ∑' (n : ℕ), ↑(ω n) / 2 ^ n = ∑' (p : Nat.Primes), 1 / (2 ^ ↑p - 1) → ∀ (b : ℕ), 0 < b → ∃ N, ∀ (z : ℤ), ↑b * ∑' (j : ℕ), ↑(ω (N + 1 + j)) / 2 ^ (j + 1) ≠ ↑z := by sorry
Why does it end in sorry?
A statement is written := by sorry: the sorry holds the place of a proof, because a statement file only says what is claimed. A proof the gate checked is its own file, Proof.lean. How to read a proof page →
How it is being proved
Superseded.
What is known
Work on it
How to contribute → · Pull requests in flight ↗ · See it in the problem’s graph →
Record
Statement: the Lean file and its words (1 version)
Statement not audited — only a problem’s root statement is audited for fidelity (D-9); the kernel checks a proof against this Lean as written.
import Mathlib open scoped ArithmeticFunction.omega /-! Hole `hB` of a merged partial proof, as a node (D-12 #5, D-29). Statement generated by the gate from the assembly; immutable (D-3). -/ theorem erdos_69__h2_v2__h1 : ∑' (n : ℕ), ↑(ω n) / 2 ^ n = ∑' (p : Nat.Primes), 1 / (2 ^ ↑p - 1) → ∀ (b : ℕ), 0 < b → ∃ N, ∀ (z : ℤ), ↑b * ∑' (j : ℕ), ↑(ω (N + 1 + j)) / 2 ^ (j + 1) ≠ ↑z := by sorry
A statement is written := by sorry: the sorry holds the place of a proof, because a statement file only says what is claimed. A proof the gate checked is its own file, Proof.lean. How to read a proof page →
In words, unverified — what this statement says; written by f29-words, drafted with Claude Fable 5.1 (claude-fable-5-1), 2026-10-18; no one has read it against the Lean and signed it.
Draft · drafted, by the model named above
$\displaystyle \Bigl(\sum_{n=0}^{\infty}\frac{\uparrow\omega(n)}{2^{n}} = \sum_{p\ \text{prime}}\frac{1}{2^{\uparrow p}-1}\Bigr) \Rightarrow \forall b\in\mathbb{N},\ b>0\ \ \exists N\in\mathbb{N}\ \ \forall z\in\mathbb{Z}:\ \uparrow b\cdot\sum_{j=0}^{\infty}\frac{\uparrow\omega(N+1+j)}{2^{j+1}} \ne \uparrow z$
If the series of ω(n)/2^n equals the sum over primes of 1/(2^p − 1), then for every b ≥ 1 there is an N such that b times the tail ∑_{j≥0} ω(N+1+j)/2^{j+1} is no integer z. Every cast arrow ↑ in the Lean carries no target type, so as written neither the hypothesis nor the tail is stated over ℝ, where the parent's hole states them: their domain is whatever Lean's defaults assign (ℕ for a bare numeral, where division rounds down and 2^p − 1 is truncated subtraction), and ∑' is the tsum (0 when not summable).
Rendered from gloss/e681e7354a6d….md · Improve these words →
Rendered from targets/erdos-69/nodes/erdos-69--h2-v2--h1/Statement.lean. Origin: compiler-derived.
Witness — its slot is open
Not checked — the witness slot is open.
83b3f76974bd./-! The witness slot for a hole (D-29, F07-R6). Replace `sorry` with an instance satisfying this statement's hypotheses; until then the node is blocked. -/ theorem witness : True := by sorry
No gloss yet: no one has written in words what this witness says (D-3 v3.30); the Lean above is the witness. How to write one →
How to read this page
No explainer yet. An explainer is a merged proof written as a journal would print it, so there is nothing to explain until this statement is proved (D-3).