Skip to content

Erdős problem 257

Start work

Let A⊆NA\subseteq\mathbb{N} be an infinite set. Is ∑n∈A12n−1 \sum_{n\in A} \frac{1}{2^n - 1} irrational?

Sources

The source's own material, retained exactly.

3 retained statements36c6d0fef71a

Open selected source

FormalConjectures/ErdosProblems/

257.lean

Retained formal statement3 of 3

Show that ∑n12n−1=∑nd(n)2n, \sum_{n} \frac{1}{2^n - 1} = \sum_{n} \frac{d(n)}{2^n}, where d(n)d(n) is the number of divisors of nn.

FormalConjectures/ErdosProblems/257.leanErdos257.erdos_257.variants.tsum_top_eq1 lineExact file
∑' (n : ℕ), 1 / (2 ^ n - 1) = ∑' (n : ℕ), ↑n.divisors.card / 2 ^ n
TextbookStatement only, no proof