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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.

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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)