FALSIFIABLE
Open, but could be disproved with a finite counterexample.
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].
View the LaTeX source
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #723, https://www.erdosproblems.com/723, accessed 2026-01-16
