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Claim. Theorem 1 of Bruck and Ryser's paper states that no finite projective plane of order $N$ exists when $N\equiv1$ or $2\pmod 4$ and the squarefree part of $N$ has a prime factor $p\equiv3\pmod 4$; equivalently, a plane of order $N\equiv1,2\pmod4$ exists only if $N$ is a sum of two integer squares. The proof turns a plane of order $N$ into a $0$-$1$ incidence matrix $A$ of order $N^2+N+1$ with $AA^{T}=A^{T}A=B$, where $B$ has $N+1$ on the diagonal and $1$ elsewhere, and applies the Hasse–Minkowski theory of rational equivalence of quadratic forms to $B$. For Problem 723 the theorem gives the answer yes for every excluded order: $6$, $14$, $21$, $22$, $30$ and every further $N\equiv1,2\pmod4$ that is not a sum of two squares has no plane, so none of these orders is a counterexample. The site records the theorem as the condition that such $n$ must be a sum of two squares and notes that it rules out $n=6$ and $n=14$.

Covers. Every order $n\equiv1$ or $2\pmod4$ that is not a sum of two squares: no plane of such an order exists, so the problem's implication holds for those $n$. It says nothing about the other orders that are not prime powers; the smallest of them, $n=10$, is excluded by Lam, Thiel and Swiercz's search, and $n=12$ is the first order the problem leaves undecided.

Acceptance. Refereed: Canad. J. Math. 1 (1949), no. 1, 88–93; the issue is dated February 1949, and the page is dated to the first day of that month. The site's curator records the theorem under [BrRy49] while labeling the problem FALSIFIABLE, which credits the partial result without settling the problem, so the page lists no reviewed evidence. The library card records the paper's theorems; the proof has not been reconstructed or independently reviewed in this corpus.