Existence of finite projective planes of non–prime-power order (especially order 12)

Determine whether a finite projective plane of order 12 exists; more generally, ascertain whether finite projective planes exist for any order that is not a prime power.

Background

Finite projective planes of order q are known to exist for all prime powers q via constructions from finite fields. Beyond prime power orders, existence is a longstanding unresolved problem in finite geometry.

The text highlights that order 12 is the smallest case for which existence is unknown, emphasizing the open status of non–prime-power orders.

References

It is unknown whether there is a projective plane of any other order. Order 12 is the smallest for which it is unknown whether a projective plane exists.

— Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof  (2510.19804 - Alexeev et al., 22 Oct 2025) in Section 4 (Cyclic projective planes)

We remark that deciding whether a finite projective plane of a given order exists is still a major open question. Although finite projective planes of prime power order exist by taking projective planes over finite fields, it is even unknown whether a finite projective plane of order $12$ exists.

— On Colorful Kruskal--Katona Theorems  (2610.02165 - Chao et al., 1 Oct 2026) in Section 1, subsection “Optimal configurations and stability”