Euler conjectured that there do not exist Euler squares of order for , 2, .... In fact, MacNeish (1921-1922) published a purported
proof of this conjecture (Bruck and Ryser 1949). While it is true that no such square
of order six exists, such squares were found to exist for all other orders
of the form by Bose, Shrikhande, and Parker in 1959 (Wells 1986, p. 77),
refuting the conjecture (and establishing unequivocally
the invalidity of MacNeish's "proof").
Bose, R. C. "On the Application of the Properties of Galois Fields to the Problem of Construction of Hyper-Graeco-Latin Squares."
Indian J. Statistics3, 323-338, 1938.Bose, R. C.;
Shrikhande, S. S.; and Parker, E. T. "Further Results on the Construction
of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture."
Canad. J. Math.12, 189, 1960.Bruck, R. H. and Ryser,
H. J. "The Nonexistence of Certain Finite Projective Planes." Canad.
J. Math.1, 88-93, 1949.Levi, F. W. Second lecture in
Finite
Geometrical Systems: Six Public Lectures Delivered in February, 1940, at the University
of Calcutta. Calcutta, India: University of Calcutta, 1942.MacNeish,
H. F. "Euler Squares." Ann. Math.23, 221-227, 1921-1922.Mann,
H. B. "On Orthogonal Latin Squares." Bull. Amer. Math. Soc.51,
185-197, 1945.Wells, D. The
Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, p. 77, 1986.
for 
, 2, .... In fact, MacNeish (1921-1922) published a purported
proof of this conjecture (Bruck and Ryser 1949). While it is true that no such square
of order six exists, such squares were found to exist for all other orders
of the form 
by Bose, Shrikhande, and Parker in 1959 (Wells 1986, p. 77),
refuting the conjecture (and establishing unequivocally
the invalidity of MacNeish's "proof").
