A Generalization of a Theorem due to MacNeish

Abstract

In 1922 MacNeish [1] considered the problem of orthogonal Latin squares and showed that if the number $s$ is written in standard form: $s = p^{n_0}_0p^{n_1}_1 \cdots p^{n_k}_k,$ where $p_0, p_1, \cdots, p_k$ are primes, and if $r = \min(p^{n_0}_0, p^{n_1}_1, \cdots, p^{n_k}_k),$ then we can construct $r - 1$ orthogonal Latin squares of side $s$. An alternative proof was also given by Mann [2]. At the April, 1950 meeting of the Institute of Mathematical Statistics at Chapel Hill, North Carolina, R. C. Bose announced an interesting generalization of this result [3] which is stated as a theorem in the next section. The proof given here is simpler than Bose's original proof and is published at his suggestion.