We shall evaluate the determinants of n x n matrices of the form [f(i,j)], where f (w, r) is an even function of m (mod r). Among the examples of determinants of this kind are H. J. S. Smith's determinant det [(i,j)], where (m, r) is the greatest common divisor of m and r, and a generalization of Smith's determinant due to T. M. Apostol.
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Paul McCarthy (1986) studied this question.