Erdős Problem #1051 — Is it true that if $a_1<a_2<\cdots$ is a sequence of integers with $\liminf a_n^{1/2^n}>1$ then $\sum_{n=1}^\infty \frac{1}{a_na_{n+1}}$ is irrational?

Erdős posed the problem with Graham in 1980, asking whether sufficiently rapid growth of an increasing integer sequence forces the reciprocal-product series to be irrational.

Known results

• Erdős, 1988: noted affirmative results when $a_n$ tends to infinity sufficiently rapidly and asked for the strongest such theorem.
• For the original condition $\liminf a_n^{1/2^n}>1$, the series is irrational by a proof using tail estimates and Mahler’s criterion.

January 2026 solution and follow-up

Aletheia, identified as a DeepMind system powered by Gemini Deep Think, produced an autonomous affirmative solution, later formalized in Lean by Barreto. Barreto, Kang, Kim, Kovač, and Zhang subsequently proved irrationality under the weaker condition $\limsup a_n^{1/\phi^n}=\infty$, with $\phi=(1+\sqrt{5})/2$, and constructed rational examples when $\lim a_n^{1/\phi^n}=C$ for every $C>1$.

Current status (as of June 2026): The original problem is settled affirmatively; the follow-up work gives a substantially sharper growth threshold and matching rational constructions.