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

The classical prime power conjecture for finite projective planes is acknowledged as an open limitation in Tancer's 2010 paper on non-representability of projective planes by convex sets: projective planes are known to exist for every prime-power order, yet no example of any other order has ever been found. The problem is open in full generality: the strongest unconditional results are the Bruck-Ryser-Chowla arithmetic obstructions, the computer-assisted exclusion of order 10, and asymptotic or computational resolutions of the abelian difference-set special case. Whether a projective plane of any non-prime-power order exists - the smallest open case being order 12 - remains unresolved.

Background

A finite projective plane of order $n \ge 2$ is a finite incidence structure of points and lines such that any two distinct points lie on exactly one common line, any two distinct lines meet in exactly one common point, and there exist four points with no three of them collinear. These axioms force the plane to have exactly $n^2+n+1$ points and $n^2+n+1$ lines, with $n+1$ points on every line; equivalently, the plane is a symmetric $2$-$(n^2+n+1, n+1, 1)$ design, meaning a collection of $n^2+n+1$ blocks (the lines), each of size $n+1$, in which every pair of points occurs in exactly one block. An integer is a prime power if it has the form $p^k$ for a prime $p$ and $k \ge 1$.

The model examples are the Desarguesian planes $PG(2, \mathbb{F}_q)$, built from the one-dimensional subspaces of a three-dimensional vector space over the finite field with $q$ elements, which exist for every prime power $q$. The existence question for other orders is a century-old problem: despite intensive study of non-Desarguesian planes (which abound at prime-power orders), every projective plane ever constructed has prime-power order, and it is widely conjectured that no others exist.

Only two kinds of unconditional negative results are known. First, the Bruck-Ryser-Chowla theorem (Bruck and Ryser 1949; Chowla and Ryser 1950) shows that if a projective plane of order $n$ exists with $n \equiv 1$ or $2 \pmod 4$, then $n$ must be expressible as a sum of two integer squares; this excludes orders such as $6, 14, 21, 22, 30, 33$, but is silent when $n \equiv 0$ or $3 \pmod 4$ and for orders that are sums of two squares, such as $18$ or $26$. Second, the nonexistence of a plane of order $10$ was settled by a massive computer search (Lam, Thiel, and Swiercz 1989), later reproduced with independently verifiable satisfiability-solver certificates (Bright et al. 2021). Beyond these, a range of conditional results constrain a hypothetical plane of order $12$ - for instance, its full collineation group (group of incidence-preserving permutations of points and lines) must be a $\{2,3\}$-group (Janko and Tran van Trung 1982), and it admits no collineation group of order $8$ (Akiyama and Suetake 2008) - without deciding existence. For the special case of planes admitting a regular abelian group of collineations, equivalently planar difference sets (subsets $D$ of a group $G$ of order $n^2+n+1$ with $|D|=n+1$ whose pairwise differences cover each non-identity element exactly once), the conjecture is essentially settled: Peluse (2021) proved an asymptotic version (the number of $n \le N$ admitting a perfect cyclic difference set is asymptotically $N/\log N$, matching the count of prime powers), and Gordon (2022) verified it computationally for all abelian difference sets of order up to $2 \cdot 10^{10}$; but an arbitrary projective plane need not carry any such abelian symmetry, so these results do not transfer.

The citing source is Tancer's paper 'Non-representability of finite projective planes by convex sets' (arXiv:0908.4038; Proc. Amer. Math. Soc. 2010), which proves that no dimension $d$ suffices to represent all finite projective planes by convex sets in $\mathbb{R}^d$. His negative results apply to the planes that are actually known to exist, and the paper states verbatim: 'It is well known that a projective plane of order $q$ exists whenever $q$ is a power of a prime. We remark that it is a well known open problem to decide whether there are projective planes of other orders.' The present problem is that classical question, not a question originating with Tancer.

Problem Statement

Does there exist a finite projective plane whose order is not a power of a prime? Equivalently, is it true that a finite projective plane of order $n$ exists if and only if $n$ is a prime power - the prime power conjecture, which asserts that the order of every finite projective plane is a prime power?

A complete answer must either (i) exhibit, for some integer $n$ that is not a prime power, an incidence structure of points and lines satisfying the projective-plane axioms (equivalently, a symmetric $2$-$(n^2+n+1, n+1, 1)$ design, or a complete set of $n-1$ mutually orthogonal Latin squares of order $n$), or (ii) prove that no such incidence structure exists for any non-prime-power $n$. The smallest undecided case is $n = 12$; resolving order 12 alone, in either direction, would be major progress but would not by itself settle the general question, since orders 15, 18, 20, 24, 26, 28, and infinitely many others would remain.

References

1. Martin Tancer, Non-representability of finite projective planes by convex sets, Proceedings of the American Mathematical Society 138 (2010) 3285-3291, DOI 10.1090/S0002-9939-10-10463-8, arXiv:0908.4038, https://doi.org/10.1090/S0002-9939-10-10463-8 (citing source: its preliminaries state the present problem as a well known open problem)
2. R. H. Bruck and H. J. Ryser, The Nonexistence of Certain Finite Projective Planes, Canadian Journal of Mathematics 1 (1949) 88-93, DOI 10.4153/CJM-1949-009-2, https://doi.org/10.4153/CJM-1949-009-2
3. S. Chowla and H. J. Ryser, Combinatorial Problems, Canadian Journal of Mathematics 2 (1950) 93-99, DOI 10.4153/cjm-1950-009-8, https://doi.org/10.4153/cjm-1950-009-8
4. C. W. H. Lam, L. Thiel, and S. Swiercz, The Non-Existence of Finite Projective Planes of Order 10, Canadian Journal of Mathematics 41 (1989) 1117-1123, DOI 10.4153/cjm-1989-049-4, https://doi.org/10.4153/cjm-1989-049-4
5. Curtis Bright, Kevin K. H. Cheung, Brett Stevens, Ilias Kotsireas, and Vijay Ganesh, A SAT-based Resolution of Lam's Problem, Proceedings of the AAAI Conference on Artificial Intelligence 35 (2021) no. 5, 3669-3676, DOI 10.1609/aaai.v35i5.16483, arXiv:2012.04715, https://doi.org/10.1609/aaai.v35i5.16483
6. Máté Matolcsi and Mihály Weiner, Character tables and the problem of existence of finite projective planes, Journal of Combinatorial Designs 26 (2018) 540-546, DOI 10.1002/jcd.21611, arXiv:1709.06149, https://doi.org/10.1002/jcd.21611
7. Sarah Peluse, An asymptotic version of the prime power conjecture for perfect difference sets, Mathematische Annalen 380 (2021), no. 3-4, 1387-1425, DOI 10.1007/s00208-021-02188-5, arXiv:2003.04929, https://doi.org/10.1007/s00208-021-02188-5
8. Daniel M. Gordon, On difference sets with small lambda, Journal of Algebraic Combinatorics 55 (2022), no. 1, 109-115, DOI 10.1007/s10801-020-00992-x, arXiv:2007.07292, https://doi.org/10.1007/s10801-020-00992-x
9. Zvonimir Janko and Tran van Trung, The full collineation group of any projective plane of order 12 is a {2,3}-group, Geometriae Dedicata 12 (1982), no. 1, 101-110, DOI 10.1007/BF00147334, https://doi.org/10.1007/BF00147334
10. Kenzi Akiyama and Chihiro Suetake, The nonexistence of projective planes of order 12 with a collineation group of order 8, Journal of Combinatorial Designs 16 (2008) no. 5, DOI 10.1002/jcd.20175, https://doi.org/10.1002/jcd.20175
11. Mingchun Xu, The properties of bordered matrix of symmetric block design, arXiv:1707.02208 (2017), https://arxiv.org/abs/1707.02208 (unrefereed preprint; claims the prime power conjecture for n <= 33, a claim not accepted in the literature)
12. Mingchun Xu, A Computer Search for a Projective Plane of Order 12, 2019 International Conference on Machine Learning and Big Data & Business Intelligence (2019), 5-8, DOI 10.1109/mlbdbi48998.2019.00009, https://doi.org/10.1109/mlbdbi48998.2019.00009