Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture

If is the prime power decomposition of an integer v , and we define the arithmetic function n(v) by then 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 as MacNeish conjectured that the actual value of N(v) is n(v).