A finite projective plane of order $n$ is a collection of subsets of $\{1,\ldots,n^2+n+1\}$ of size $n+1$ such that every pair of elements is contained in exactly one set.

OPEN

If there is a finite projective plane of order $n$ then must $n$ be a prime power?

A finite projective plane of order $n$ is a collection of subsets of $\{1,\ldots,n^2+n+1\}$ of size $n+1$ such that every pair of elements is contained in exactly one set.

These always exist if $n$ is a prime power. This conjecture has been proved for $n\leq 11$, but it is open whether there exists a projective plane of order $12$.

Bruck and Ryser [BrRy49] have proved that if $n\equiv 1\pmod{4}$ or $n\equiv 2\pmod{4}$ then $n$ must be the sum of two squares. For example, this rules out $n=6$ or $n=14$. The case $n=10$ was ruled out by computer search [La97].