Are the nth roots of the primes strictly decreasing? Firoozbakht's 1982 guess would control prime gaps far beyond Legendre — verified computationally, proved nowhere.
The Problems
Not schoolwork — the questions that resisted Erdős, Hilbert, and everyone since. The club keeps three shelves: what is still open, what AI recently settled, and what took humanity centuries.
Tagged prime-numbers — 3 entries. Clear
The Riesel problem
open
The distributed-computing classic: is 509203 the smallest k for which k·2ⁿ − 1 is never prime? PrimeGrid grinds on; the answer is a finite search away.
Iterated absolute differences of the primes always start with 1. Verified to 10^13 and beyond, proved nowhere — perhaps the most elementary open problem about the primes.