Is $\sum_n p_n/2^n$ irrational, where $p_n$ is the $n$th prime? (Erdős #251)
Statement
Let $p_n$ denote the $n$th prime ($p_1=2,\ p_2=3,\ p_3=5,\dots$). Is the real number $$\sum_{n=1}^{\infty}\frac{p_n}{2^n}$$ irrational? The series converges (since $p_n=O(n\log n)$ grows far slower than $2^n$), and the question is whether its value is rational.
Acceptance. FULLY RESOLVES: a complete rigorous proof that $\sum_n p_n/2^n$ is irrational, OR a proof that it is rational together with its exact value — machine-checkable Lean/Coq preferred (a formal statement exists in DeepMind's formal-conjectures repo), otherwise a full written proof with all steps explicit. Digit computation alone can NEVER close it. ADVANCES (each itself checkable): an effective irrationality measure for the value; a proof ruling out all rational values with denominator below an explicit bound; an irrationality proof of a related sum in the same family that is strictly new relative to the background (which records only that $\sum_n p_n^k/n!$ is irrational, Erdős [Er58b]) — e.g. settling $\sum_n p_n^k/2^n$ for some $k$, or $\sum_n p_n/(g_1\cdots g_n)$ for an explicit admissible sequence $g_n$; or an irrationality proof conditional on a clearly-stated standard conjecture on prime gaps. State precisely which sum you resolve and under what hypotheses. Deliver the Lean proof file or a complete written proof.
Background
Asked by Erdős [Er58b] and repeated in Erdős–Graham [ErGr80, p.62] and Erdős [Er88c, p.103]; listed as open on erdosproblems.com/251 (fetched 2026-07-13, status 'open', tagged 'number theory | irrationality'), with no prize. The decimal expansion is OEIS A098990, and a formal statement exists in DeepMind's formal-conjectures Lean library. What IS known is a factorial-denominator analogue: Erdős [Er58b] proved that $\sum_n p_n^k/n!$ is irrational for every integer $k\ge1$ — the faster factorial decay makes the classical irrationality argument go through. The $2^n$-denominator version resisted that method. In [Er88c] Erdős conjectured more broadly that $\sum_n p_n^k/2^n$ is irrational for every $k$, and that if $g_n\ge2$ with $g_n=o(p_n)$ then $\sum_{n\ge1} p_n/(g_1\cdots g_n)$ is irrational — the choice $g_n=p_n+1$, which makes the sum telescope to a rational, shows that some growth restriction on the $g_n$ is necessary. This problem is the base case $k=1$, $g_n=2$. Irrationality being a non-finite property, digit computation cannot resolve it; the attacker's tool is analytic irrationality machinery — denominator/gcd growth on partial sums exploiting prime-gap regularity, or a Lean formalisation of a new criterion — with extension of A098990 giving only heuristic support.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #251 (T. F. Bloom) | website |
| REF-02 | OEIS A098990 — decimal expansion of sum p_n/2^n | website |
| REF-03 | DeepMind formal-conjectures — Lean statement of Erdős #251 | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.