The nonexistence of certain finite projective planes

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theorem_1: Bruck and Ryser's theorem that no finite projective plane with N + 1 points on a line exists when N is congruent to 1 or 2 mod 4 and the squarefree part of N has a prime factor of the form 4k + 3, with the paper's remark that no complete set of mutually orthogonal Latin squares of such an order N exists.

theorem_2: Bruck and Ryser's theorem that a finite projective plane with N + 1 points on a line has an incidence matrix A of order N^2 + N + 1 with AA^T = A^TA = B, where B has N + 1 on the diagonal and ones elsewhere.

theorem_3: Bruck and Ryser's converse to their Theorem 2: a matrix of order n > 1 with nonnegative integral entries satisfying (M) with N at least 2 is an incidence matrix and defines a finite projective plane with N + 1 points on a line.

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Bruck, R. H. and Ryser, H. J., The nonexistence of certain finite projective planes. Canad. J. Math. 1 (1949), 88-93.

Theorem 1 (p. 88) is the Bruck-Ryser theorem: if N = 1 or 2 mod 4 and the squarefree part of N contains at least one prime factor of the form 4k+3, then there is no finite projective plane with N+1 points on a line. Theorem 2 (p. 89) turns such a plane into a 0-1 incidence matrix A of order n = N^2+N+1 with AA^T = A^TA = B, where B has N+1 on the diagonal and ones elsewhere (equation (M)); Theorem 3 (p. 89) is the converse for nonnegative integral solutions of (M) with N >= 2. Section 3 (pp. 89--91) recalls the Hilbert norm-residue symbol (Theorems 4 and 5, p. 90, cited from Hilbert), proves a Lemma on it (p. 90) and states the Minkowski-Hasse theorem on rational congruence of quadratic forms (Theorem 6, p. 91, cited from Hasse). Section 4 (pp. 91--92) computes the Hasse-Minkowski invariant of B and derives the contradiction. The authors note (p. 88) that Theorem 1 rules out in particular the planes of order N = 2p with p a prime of the form 4k+3, and that, since a complete set of mutually orthogonal Latin squares of order N >= 3 gives a plane, no such complete set exists for any N of Theorem 1. A postscript (pp. 92--93) records Marshall Hall's simpler route to the key equation (E) and comments on Euler's conjecture on orthogonal Latin squares of order 4k+2.

Read status: claims checked for Theorems 1, 2 and 3 and the remarks on p. 88 (read clause by clause on the page images of the print); the proof of Theorem 1 was followed in outline, and Theorems 4 to 6, which the paper cites, were not checked. Nothing here is independently reviewed.

Result pages

• Theorem 1 (p. 88): the nonexistence theorem, with the paper's remarks on orders $2p$ and on complete sets of orthogonal Latin squares, and Postscript (b).
• Theorem 2 (p. 89): a plane gives an incidence matrix satisfying (M).
• Theorem 3 (p. 89): a nonnegative integral solution of (M) with $N\ge2$ defines a plane.

Theorems 4 to 6 and the Lemma (pp. 90--91) are background on quadratic forms used in the proof and have no pages of their own.

Source: https://doi.org/10.4153/cjm-1949-009-2. No notice is printed (the running footer "Published online by Cambridge University Press" is not one); the journal's article page on Cambridge Core shows "Copyright © Canadian Mathematical Society 1949" and names no Creative Commons license (https://www.cambridge.org/core/product/identifier/S0008414X00028686/type/journal_article, read 2026-10-02), every other right reserved.

Bears on. #723, whether every finite projective plane has prime-power order:

• Theorem 1 excludes every order $N\equiv1,2\pmod4$ whose squarefree part has a prime factor $\equiv3\pmod4$ (equivalently, that is not a sum of two squares), among them $6$, $14$, $21$ and $22$; no prime power is among them. It excludes no order $\equiv0,3\pmod4$, such as $12$, and no sum of two squares, such as $10$, so it does not settle the problem.
• Theorem 2 and Theorem 3 restate, for $N\ge2$, the existence of a plane of order $N$ as the existence of a nonnegative integral matrix solution of (M); they exclude no order.

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