Originally Posted by Jameson
Thus, we have,
Now multiply (2)(3)(4) to get,
The reason being that,
and all thus, they are not squares. Then, .
Here is an outline of the proof, to keep thing simple let us do it with only two terms,
and are positive integers with a non-square.
Now if then,
Impossible because LHS is irrational.
If you have, you get the same thing.
Thus, which implies that