Erdős Problem #251

The problem

Is

$$\sum \frac{p_n}{2^n}$$

irrational? (Here $p_n$ is the $n$th prime.)

Statement quoted from erdosproblems.com/251, maintained by Thomas Bloom. Original source: [Er58b][ErGr80,p.62][Er88c,p.103]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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