Existence of finite projective planes of non-prime-power order (the prime power conjecture)

The existence problem for finite projective planes asks for which integers $n\ge 2$ there exists a projective (equivalently, affine) plane of order $n$. Planes exist at every prime-power order via the Desarguesian construction $PG(2,q)$ over the finite field with $q$ elements, and no plane of any other order has ever been found; besides the infinitely many orders excluded in general by the Bruck–Ryser–Chowla theorem, nonexistence has been settled by arguments beyond that criterion only for orders 6 and 10, the latter by a landmark exhaustive computation. The prime power conjecture asserts that prime-power orders are the only possible ones. It remains open: the smallest unresolved case is order 12, and the standard general tools (Bruck–Ryser–Chowla conditions, linear-programming and character-theoretic bounds, $p$-rank inequalities, conditional collineation-group exclusions) do not settle it.

Background

A finite projective plane of order $n$ is an incidence structure of points and lines such that every line contains exactly $n+1$ points, every point lies on exactly $n+1$ lines, and any two distinct points lie on exactly one common line; it follows that the plane has exactly $n^2+n+1$ points and equally many lines, and, dually, that any two distinct lines meet in exactly one point. An affine plane of order $n$ is what remains after deleting one line and all of its points from a projective plane, and conversely every affine plane extends to a projective plane by adjoining a line at infinity, so the existence of an affine plane of order $n$ is equivalent to that of a projective plane of order $n$. Equivalently again, a projective plane of order $n$ exists precisely when there is a complete set of $n-1$ mutually orthogonal Latin squares of order $n$ (a Latin square is an $n\times n$ array in which each symbol occurs exactly once in every row and column; two Latin squares are orthogonal if, superimposed, they produce every ordered pair of symbols exactly once).

When $q$ is a prime power, the finite field $GF(q)$ with $q$ elements yields the Desarguesian plane $PG(2,q)$, coordinatized by homogeneous coordinates over $GF(q)$, and every finite projective plane known today has prime-power order. The classical general obstruction is the Bruck–Ryser–Chowla theorem: if $n\equiv 1$ or $2\pmod{4}$, then $n$ must be a sum of two squares, which excludes infinitely many orders ($6, 14, 21, 22, 30,\dots$) but says nothing about $n\equiv 0$ or $3\pmod{4}$ (such as $12, 15, 20, 24, 28$) nor about sums of two squares like $18=9+9$ or $26=25+1$. Beyond this, only two individual orders have been settled: order $6$, already excluded by Bruck–Ryser–Chowla and by early exhaustive enumeration, and order $10$, whose nonexistence was established in 1989 by Lam, Thiel, and Swiercz through one of the largest exhaustive computer searches ever completed in combinatorics. The smallest order whose status is unknown is $n=12$.

Bruen's 2006 survey of applications of finite fields to combinatorics and finite geometries lists this problem among the open directions of the area and proves a lower bound $\mathrm{Rank}_p(M)\ge n\sqrt{n}+1$ on the $p$-rank — the rank over the finite field $GF(p)$ — of the point–line incidence matrix $M$ of a projective plane of order $n$, for any prime $p$ dividing $n$; such algebraic constraints on incidence matrices are among the standard tools used to attack nonexistence. It is widely conjectured — the prime power conjecture for finite projective planes — that planes exist only at prime-power orders. The question predates the survey by decades and stands as one of the oldest open problems of finite geometry and design theory.

Problem Statement

Determine for which integers $n\ge 2$ there exists a finite projective plane of order $n$ — equivalently, an affine plane of order $n$, or a complete set of $n-1$ mutually orthogonal Latin squares of order $n$. Since planes exist at every prime-power order, the open content of the problem is exactly the non-prime-power orders: does there exist a projective plane of order $n$ for any integer $n$ with at least two distinct prime factors, the smallest such case being $n=12$? Equivalently, is the prime power conjecture true: does a projective plane of order $n$ exist if and only if $n$ is a prime power? The problem is resolved only by a construction of a plane at some non-prime-power order (which would refute the conjecture) or by a proof that no projective plane exists at any non-prime-power order (equivalently, a proof of the prime power conjecture); a nonexistence proof for a single specific order such as 12 would constitute significant partial progress but would not by itself resolve the problem.

References

1. Bruen, A. A.: Applications of Finite Fields to Combinatorics and Finite Geometries. Acta Applicandae Mathematicae 93(1-3), 179-196 (2006). DOI: 10.1007/s10440-006-9051-4. https://doi.org/10.1007/s10440-006-9051-4
2. Bruck, R. H., Ryser, H. J.: The Nonexistence of Certain Finite Projective Planes. Canadian Journal of Mathematics 1, 88-93 (1949). DOI: 10.4153/CJM-1949-009-2. https://doi.org/10.4153/CJM-1949-009-2
3. Chowla, S., Ryser, H. J.: Combinatorial Problems. Canadian Journal of Mathematics 2, 93-99 (1950). DOI: 10.4153/cjm-1950-009-8. https://doi.org/10.4153/cjm-1950-009-8
4. Lam, C. W. H., Thiel, L., Swiercz, S.: The Non-Existence of Finite Projective Planes of Order 10. Canadian Journal of Mathematics 41(6), 1117-1123 (1989). DOI: 10.4153/cjm-1989-049-4. https://doi.org/10.4153/cjm-1989-049-4
5. Akiyama, K., Suetake, C.: The nonexistence of projective planes of order 12 with a collineation group of order 8. Journal of Combinatorial Designs 16(5), 411-430 (2008; published online November 2007). DOI: 10.1002/jcd.20175. https://doi.org/10.1002/jcd.20175
6. Akiyama, K., Suetake, C., Tanaka, M.: Projective planes of order 12 do not have a collineation group of order 4. Journal of Combinatorial Designs 31(2), 87-123 (2023; published online December 2022). DOI: 10.1002/jcd.21869. https://doi.org/10.1002/jcd.21869
7. Matolcsi, M., Weiner, M.: Character tables and the problem of existence of finite projective planes. Journal of Combinatorial Designs 26, 540-546 (2018). DOI: 10.1002/jcd.21611. https://doi.org/10.1002/jcd.21611
8. Matolcsi, M., Weiner, M.: Finite Projective Planes and the Delsarte LP-Bound. Analysis Mathematica 44, 89-98 (2018). DOI: 10.1007/s10476-018-0108-1. https://doi.org/10.1007/s10476-018-0108-1
9. Xu, M.: The properties of bordered matrix of symmetric block design. arXiv:1707.02208 (2017). https://arxiv.org/abs/1707.02208