2122.15 – Pairs of Integers

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

m3+6m2+5m=27n3+9n2+9n+1m^3 + 6m^2 + 5m = 27n^3 + 9n^2 + 9n + 1

Solution

This looks horrible. But take a good look at the nn expression. It can be written 9(3n3+n2+n)+19(3n^3 + n^2 + n) + 1, which might be useful.

What about factoring the mm expression?

m3+6m2+5m=m(m2+6m+5)=m(m+1)(m+5)\begin{align*}m^3 + 6m^2 + 5m &= m(m^2 + 6m + 5) \\&= m(m+1)(m+5)\end{align*}

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

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