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-gaps — 6 entries. Clear
Each half of the interval between consecutive squares contains a prime. Stronger than Legendre's conjecture, from which it directly implies the one-prime-per-square case.
Andrica's Conjecture
open
The square roots of consecutive primes always differ by less than one. A prime-gap conjecture strictly stronger than Legendre's, open since 1986.
There is always a prime between consecutive perfect squares. The oldest open problem about prime gaps — weaker than Oppermann's, Andrica's and Cramér's conjectures, and implied by each of them.
Cramér's Conjecture
open
How far apart can consecutive primes be? Cramér's probabilistic model says never much more than (log p)² — a prediction we cannot approach unconditionally.
There are infinitely many primes p such that p + 2 is also prime. Infinitely close, structurally — but unproven.