A determinant (2)

Number theory
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Tolaso J Kos
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A determinant (2)

#1

Post by Tolaso J Kos »

Let $[a, b]$ denote the least common multiple of $a, b$. Prove that:

$$\begin{vmatrix}
\left [ 1,1 \right ] &\left [ 1,2 \right ] &\cdots &\left [1,n \right ] \\
\left [ 2,1 \right ]&\left [ 2,2 \right ] & \cdots &\left [ 2,n \right ] \\
\vdots & \vdots & \ddots & \vdots \\
\left [ n,1 \right ]& \left [ n, 2 \right ] &\cdots & \left [ n,n \right ]
\end{vmatrix} = \prod_{k=1}^{n}\phi(k) \prod_{p \mid k} (-p)$$

where $\phi$ denotes Euler's $\phi$ function.
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