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On the Maximal Number of Pairwise Orthogonal Latin Squares of a Given Order

Published online by Cambridge University Press:  20 November 2018

S. Chowla ,
P. Erdös  and
E. G. Straus
S. Chowla
Affiliation:
Number Theory Institute (1959), BoulderColorado
P. Erdös
Affiliation:
University of Colorado
E. G. Straus
Affiliation:
University of California, Los Angeles
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In the preceding paper Bose, Shrikhande, and Parker give their important discovery of the disproof of Euler's conjecture on Latin squares. In this paper we show that their results can be strengthened to imply that N(n), the maximal number of pairwise orthogonal Latin squares of order n, tends to infinity with n. In fact there exists a positive constant c, such that N(n) > nc for all sufficiently large n.

Our proof involves no new combinatorial insights, but is based entirely on a number-theoretical investigation of the following inequality due to Bose and Shrikhande.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 204 - 208
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Rademacher, H., Beitràge zur Viggo Brunschen Méthode in der Zahlentheorie, Abbh. Math. Sem. Hamburg, 3 (1924), 12–30.Google Scholar