Let
be the
matrix whose th
entry is 1 if
divides
and 0 otherwise, let
be the diagonal matrix , where is the totient function,
and let
be the
matrix whose th
entry is the greatest common divisor . Then Le Paige's theorem states
that
where
denotes the transpose (Le Paige 1878, Johnson 2003).
As a corollary,
(Smith 1876, Johnson 2003). For , 2, ... the first few values are 1, 1, 2, 4, 16, 32, 192,
768, ... (OEIS A001088).
Johnson, W. P. "An Factorization in Elementary Number Theory." Math.
Mag.76, 392-394, 2003.Le Paige, C. "Sur un théorème
de M. Mansion." Nouv. Corresp. Math.4, 176-178, 1878.Mansion,
P. "On an Arithmetical Theorem of Professor Smith's." Messenger Math.7,
81-82, 1877.Muir, T. A
Treatise on the Theory of Determinants, Vol. 3. New York: Dover, 1960.Sloane,
N. J. A. Sequence A001088 in "The
On-Line Encyclopedia of Integer Sequences."Smith, H. J. S.
"On the Value of a Certain Arithmetical Determinant." Proc. London Math.
Soc.7, 208-212, 1876. Reprinted in The Collected Mathematical Papers
of Henry John Stephen Smith, Vol. 2 (Ed. J. W. L. Glaisher).
Oxford, England: Clarendon Press, pp. 161-165, 1894.