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  4. Smith's Determinant

Smith's Determinant

John Cook user avatar by
John Cook
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Aug. 01, 13 · Interview
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here’s a curious identity i ran across while browsing knuth’s magnum opus .

\[\left|    \begin{array}{llll}       \gcd(1,1) & \gcd(1,2) & \ldots & \gcd(1, n) \\       \gcd(2,1) & \gcd(2,2) & \ldots & \gcd(2, n) \\       \vdots & \vdots & & \vdots \\       \gcd(n,1) & \gcd(n,2) & \ldots & \gcd(n, n)     \end{array}\right| = \varphi(1) \varphi(2) \cdots \varphi(n) \>

here gcd( i , j ) is the greatest common divisor of i and j , and φ( n ) is euler’s phi function, the number of positive integers less than or equal to n and relatively prime to n .

the more general version of smith’s determinant replaces gcd( i , j ) above with f ( gcd( i , j ) ) for an arbitrary function f .

as a hint at a proof, you can do column operations on the matrix to obtain a triangular matrix with φ( i ) in the i th spot along the diagonal.

source: taocp vol 2, section 4.5.2, problem 42.

Matrix (protocol) PRIME (PLC) Column (database) Euler (software)

Published at DZone with permission of John Cook, DZone MVB. See the original article here.

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