Problem 723

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.

Source: erdosproblems.com/723

Combinatorics

Wiki page

Status

Open

No claim settles this problem.

Falsifiable: the site labels the problem FALSIFIABLE and records that planes exist for every prime-power order, that the conjecture holds for $n\le11$ while the existence of a plane of order $12$ is open, Bruck and Ryser's theorem [BrRy49] that an order $n\equiv1$ or $2\pmod4$ must be a sum of two squares, which rules out $n=6$ and $n=14$ , and the computer search that ruled out $n=10$ [La97].

BrRy49: Bruck, R. H. and Ryser, H. J., The nonexistence of certain finite projective planes. Canad. J. Math. 1 (1949), 88-93.
La97: Lam, C. W. H., The search for a finite projective plane of order $10$ . Amer. Math. Monthly 98 (1991), no. 4, 305--318 [MR1103185 (92b:51013)]; the site's entry, "(1997), 335-355", refers to a reprint of the Monthly article, not held in this corpus. Library home: Lam 1997 (the author's 2005 revision of the article).
LTS89: Lam, C. W. H., Thiel, L. and Swiercz, S., The non-existence of finite projective planes of order 10. Canad. J. Math. 41 (1989), no. 6, 1117–1123. Not in the site's bibliography; the research paper behind the search [La97] reports. Not held.

Problem 723

Status

Proof claims

Discussion