Thèse de doctorat de 3e cycle (Mathématiques Statistiques),
Faculté des Sciences, Paris, 1959.
Dominique Foata
Sur quelques applications du problème d'Euler-Tarry à l'analyse de la variance
Abstract.
Recall that a Latin square of order n is a square matrix, whose rows and columns are permutations of the sequence 1 2 3 ... n . Two Latin squares are said to be orthogonal if when one is superimposed upon the other every ordered pair of integers 11,12, ... , (n-1)n, nn occurs once in the resulting square. Euler conjectured that there did not exist any pair of orthogonal Latin squares of order 4m+2. He was correct for the integer m=1, as was proved by the French General Tarry (1900), but hopelessly wrong (as Bose used to say) for every m greater than or equal to 2, as proved by Bose, R. C., Shrikhande, S. S. and Parker E. T. (1959). Both Latin squares and orthogonal pairs had been used as designs of experiments in the Analysis of Variance. The purpose of the paper was to make a parallel study for the so-called quasi-orthogonal Latin squares, as it was not yet known that true orthogonal ones did exist except for n=2 and 6.
No archive available, as the paper has yellowed and became unreadable.
foata@unistra.fr