For any positive integer $n$, $i,j,k$ are also positive integers, and $0 . How many solutions of the form $(i,j,k)$ are there for the equation $ijk = (n-i)(n-j)(n-k)$?
Is there some exact form of $n$ for the number of $(i,j,k)$ satisfying $ijk = (n-i)(n-j)(n-k)$
8
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number-theory
elementary-number-theory
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0If $i=j=k$ then $(2i,i,i,i)$ is a solution for all $i$. Set $n=i+j$ we get the another solution set $(i+j, i, j,\frac{i+j}{2})$. Replacing even $i$ & $j$ we get one set $(2i+2j, 2i, 2j, i+j)$. For both odd we get another $(2i+2j+2, 2i+1,2j+1, i+j+1)$ – 2015-07-11