Le Paige's Theorem

Let L_n be the n×n matrix whose (i,j)th entry is 1 if j divides i and 0 otherwise, let Phi_n be the n×n diagonal matrix diag(phi(1),phi(2),...,phi(n)), where phi(n) is the totient function, and let G_n be the n×n matrix whose (i,j)th entry is the greatest common divisor GCD(i,j). Then Le Paige's theorem states that


where A^(T) denotes the transpose (Le Paige 1878, Johnson 2003).

As a corollary,


(Smith 1876, Johnson 2003). For n=1, 2, ... the first few values are 1, 1, 2, 4, 16, 32, 192, 768, ... (OEIS A001088).

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Johnson, W. P. "An LDU 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.

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Le Paige's Theorem

Cite this as:

Weisstein, Eric W. "Le Paige's Theorem." From MathWorld--A Wolfram Web Resource.

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