The prime-power conjecture for finite projective planes

Let a finite projective plane of order $n$ be a symmetric design with parameters $(n^2+n+1,n+1,1)$.

Prime-power conjecture. If a finite projective plane of order $n$ exists, then $n$ is a power of some prime $p$.

Projective planes are known to exist for every prime-power order, while no construction is known for other orders. The conjecture is a long-standing open problem in finite geometry; the Bruck–Ryser–Chowla theorem rules out some non-prime-power orders, but does not settle the general case.