On the construction of sets of mutually orthogonal Latin squares and the falsity of a conjecture of Euler

Abstract

then it is known, MacNeish [16] and Mann [17] that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. It seemed plausible that n(v) is also the maximum possible number of m.o.l.s. of order v. This would have implied the correctness of Euler's [13] conjecture about the nonexistence of two orthogonal Latin squares of order v when v =2 (mod 4), since n(v) -1 in this case. If we denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v, then Parker [18 ] showed that in certain cases N(v) >n(v) by proving that if there exists a balanced incomplete block (BIB) design with v treatments, X = 1, and block size k which is a prime power then N(v) > k -2. This result cast a doubt on the validity of Euler's conjecture. In this paper we generalize Parker's method of constructing m.o.l.s., by showing that a very general class of designs, which we have called pairwise balanced designs of index unity, can be used for the construction of sets of m.o.l.s. Various applications of this method have been made. In particular we show that Euler's conjecture is false for an infinity of values of v, including all values of the form 36m+22. We also provide a table for all values of v n(v). The smallest number for which we have been able to demonstrate the falsity of Euler's conjecture is 22. Two orthogonal squares of order 22 constructed by our method are given in the Appendix. Several attempts, necessarily erroneous, have been made in the past to prove Euler's conjecture, e.g., Peterson [19], Wernicke [20] and MacNeish [16]. Levi [14, p. 14] points