2122.15 – Pairs of Integers

Find all the pairs (m,n) of integers which satisfy the equation

$m^3 + 6m^2 + 5m = 27n^3 + 9n^2 + 9n + 1$ .

Solution

This looks horrible. But take a good look at the $n$ expression. It can be written $9(3n^3 + n^2 + n) + 1$ , which might be useful.

What about factoring the $m$ expression? $m^3 + 6m^2 + 5m = m(m^2 + 6m + 5)$ , ha, = $m(m+1)(m+5)$ .

Ponder this: the $m$ expression will always have a factor of 3, no matter what $m$ itself is (you might have to think about this for a minute). And, from what you did with the $n$ expression, you see that it never has a factor of 3, no matter what $n$ is.

So there can't be any integers that satisfy the entire equation.