A Generalization of a Theorem due to MacNeish

In 1922 MacNeish [1] considered the problem of orthogonal Latin squares and showed that if the number s is written in standard form: s = pn₀₀pn₁₁ ⋯ pnₖₖ, where p₀, p₁, ⋯, pₖ are primes, and if r = min(pn₀₀, pn₁₁, ⋯, pnₖₖ), 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.