Ifis the prime power decomposition of an integer v, and we define the arithmetic function n(v) bythen it is known, MacNeish (10) and Mann (11), that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. We shall denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v. Then the Mann-MacNeish theorem can be stated asMacNeish conjectured that the actual value of N(v) is n(v).

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y in (q2 n ... n q)-p we have that (xj): y = p. Since x; is not in Up and hd p < 1, it follows from the previous proposition that p = (xj). THEOREM 3. Let R be a local domain of dimension 1 and dim R < 3. Thus hd R/p < 2, which completes the proof. Since every module has finite homological dimension over a regular local ring, we have established COROLLARY 4. Every regular local ring of dimension <3 is a unique factorization domain. THEOREM 5. Every regular local ring is a unique factorization domain. * Prior to this result, Zariski proved that if every complete regular local ring of dimension 3 is a unique factorization domain, then every complete regular local ring is a unique factorization domain (unpublished). Combining this with Mori's and Krull's result that a local ring is a unique factorization domain if it's completion is a unique factorization domain, we obtain another proof of this reduction theorem.