Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 7

Problem 723

claims/: The 2 claim pages of Problem 723, one per claimant's result; the problem's standing derives from them.

Statement. 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.

Status. 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].

Source. erdosproblems.com/723, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #723, https://www.erdosproblems.com/723.

References.

[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_search_finite_projective_plane_order_10 (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.

Formalization. Statement in formal-conjectures, left unproved there with the variants it lists, the order- $12$ question tagged open and the others tagged solved; it names no formal proof.

Current assessment

The question asks whether every finite projective plane has prime-power order. The standing is open: no result settles or claims to settle the question, and the two claim pages are accepted partial claims on refereed evidence. The site labels the problem falsifiable, since a counterexample is one finite projective plane of an order that is not a prime power, a finite set system whose defining property can be checked by enumeration; this is a body note, not a claim. Bruck and Ryser's theorem [BrRy49] excludes every order $n\equiv1$ or $2\pmod4$ that is not a sum of two squares, among them $6$ , $14$ , $21$ and $22$ ; Lam, Thiel and Swiercz's computer search [LTS89], which the site cites through Lam's expository article [La97], excludes order $10$ . Planes exist for every prime-power order, the converse direction, which is not a claim about the question. With these the conjecture holds for $n\le11$ , and order $12$ is the first undecided case. The formal-conjectures statement file, at the commit linked above, leaves the problem, its open variant asking whether a plane of order $12$ exists, and the solved variants it lists (prime-power orders, orders at most $11$ , the Bruck–Ryser condition) unproved and names no formal proof. Nothing was reconstructed in this corpus.

Search scope, 2026-10-07: the site's problem page and discussion thread, which record no proof claim, the community database entry, the formal-conjectures statement file, the Crossref records of [BrRy49] and [LTS89], and the two library cards.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (14)

Problem 665 Shrikhande and Singhi's embedding theorem, a conditional no Bruck and Ryser's exclusion of orders that are not sums of two squares Lam, Thiel and Swiercz's nonexistence of a plane of order 10 problems/set_systems/E0723/claims Set Systems, Designs and Hypergraphs The nonexistence of certain finite projective planes Theorem 1 (p. 88): no finite projective plane of order N ≡ 1, 2 (mod 4) whose squarefree part has a prime factor 4k + 3 Theorem 2 (p. 89): a plane with N + 1 points on a line gives an incidence matrix A of order N^2 + N + 1 with AA^T = A^TA = B Theorem 3 (p. 89): a nonnegative integral solution of (M) with N ≥ 2 is an incidence matrix of a plane with N + 1 points on a line set_systems/lam_1997_search_finite_projective_plane_order_10 Reported result (pp. 8--19): the computer search found no finite projective plane of order 10 Theorem 2 (p. 4): for n ≥ 3, a plane of order n exists iff a complete set of n − 1 MOLS of order n exists Theorem 3 (p. 5): Bruck and Ryser's condition that a plane of order n ≡ 1, 2 (mod 4) needs n = x^2 + y^2

Graph