Is $\sum_{n\in A}1/(2^n-1)$ irrational for every infinite set $A\subseteq\mathbb{N}$? (Erdős #257)
Statement
Let $A\subseteq\mathbb{N}$ be an arbitrary infinite set of positive integers. Is the sum $$\sum_{n\in A}\frac{1}{2^n-1}$$ always irrational? Equivalently, since $\frac{1}{2^n-1}=\sum_{k\ge 1}2^{-kn}$, writing $f_A(m)$ for the number of divisors of $m$ that lie in $A$, one asks whether $\sum_{m\ge 1} f_A(m)\,2^{-m}$ is irrational for every infinite $A$.
Acceptance. FULLY RESOLVES: a complete proof — machine-checkable in Lean/Coq preferred, since a formalisation stub already exists, else a full written proof with all steps — that $\sum_{n\in A}1/(2^n-1)$ is irrational for every infinite $A\subseteq\mathbb{N}$; OR a counterexample, namely an explicit infinite set $A$ for which the sum is rational, together with a proof of rationality (e.g. an exact closed-form value). ADVANCES (each independently checkable): prove irrationality for a strictly larger class of sets $A$ than the best stated in the background (currently pairwise-coprime $A$ with $\sum_{n\in A}1/n<\infty$, plus the primes and prime powers) — for instance dropping the coprimality hypothesis, or covering all $A$ of positive lower density — with proof; OR prove the base-$t$ generalisation for some integer $t\ge 3$; OR complete the existing Lean formalisation for a nontrivial sub-case. Deliver the proof file, the new-class proof, or the rational counterexample with its certified value.
Background
Posed by Erdős [Er68d], [ErGr80, p.62], [Er88c, p.105]. Frontier: for $A=\mathbb{N}$ the sum equals $\sum_m \tau(m)2^{-m}$ with $\tau$ the divisor-counting function, which Erdős [Er48] proved irrational. Erdős [Er68d] proved irrationality whenever $A$ is pairwise coprime and $\sum_{n\in A}1/n<\infty$ (he believed the coprimality hypothesis could be removed by complicating the proof). The case $A=$ the primes (erdosproblems #69), and the case of prime powers, were settled affirmatively by Tao and Teräväinen [TaTe25]. There is nothing special about the base $2$: the analogue with $2$ replaced by any integer $t\ge 2$ is expected to hold. A broader speculation of Erdős [Er88c] — that $\sum_{n\in A}1/(2^n-t_n)$ is irrational for every infinite $A$ and every bounded integer sequence $t_n$ — was disproved by Kovač and Tao [KoTa24], who (per Kovač's site comment) exhibit a sequence with $1\le t_n\le 6$ making the sum rational; crucially this does NOT touch the constant case $t_n\equiv 1$ of #257, which remains open. A Lean formalisation of the statement already exists in the google-deepmind/formal-conjectures repository. Attacker's tool: analytic irrationality machinery (Erdős-style divisor-sum arguments and the Tao–Teräväinen circle of methods that settled the primes case), or completion of the existing Lean formalisation; numerical continued-fraction experiments can only rule out simple rational values for specific families $A$, not prove the general statement. Listed as open on erdosproblems.com/257 (fetched 2026-07-13, status 'open').
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #257 (T. F. Bloom) | website |
| REF-02 | Lean formalisation — google-deepmind/formal-conjectures, ErdosProblems/257.lean | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.