Erdős Problem #521
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`true` encodes the sign `+1`, `false` the sign `-1`.
One fair coin: the uniform probability measure on `Bool`.
The law of an infinite sequence of independent fair coins `(ε_k)_{k ≥ 0}`, each uniform on `{-1, +1}`.
Let \((\epsilon_k)_{k\geq 0}\) be independently uniformly chosen at random from \(\{-1,1\}\). If \(R_n\) counts the number of real roots of \(f_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k\) then is it true that, almost surely, \[\lim_{n\to \infty}\frac{R_n}{\log n}=\frac{2}{\pi}?\] The answer is no: this almost-sure limit fails. This result was obtained first by others, who deserve the credit for the problem; the link is to an independent machine-checked proof by Star Fleet Math.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/521.Prizenone
Topicsanalysis, polynomials, probability
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/521
Attempts
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Live page status OPEN, but it lists 1 claimed proof (a full proof claimed by Rob Sneiderman, submitted 2026-07-21) and identifies lof310 as currently working on the problem.
SKIPPED