# Positive Integers Equation

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with the least common multiple, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it. So anyway, here the problem goes:

Find all solutions in positive integers for the equation $$x-y^4= LCM(x, y).$$

Any subtantial answers or help would be truly greatly appreciated. Thanks so much :)

• What is $LCM(1)$ supposed to mean...? Commented Jul 23, 2015 at 21:56
• You have $\operatorname{lcm}(x,1) = x$, so setting $y = 1$ doesn't work. Commented Jul 23, 2015 at 21:56
• @DanielFischer I am thoroughly confused. I don't know how to do it. Commented Jul 23, 2015 at 21:58
• You must have $x \mid x - y^4$ and $y \mid x-y^4$. That gives you some conditions to work with. Commented Jul 23, 2015 at 21:58
• @DanielFischer Sorry I don't understand. Could you please explain in answer? Commented Jul 23, 2015 at 22:01

AFAIK the least multiple of any positive integer $x$ is $x$ itself. As $y^4$ is greater than zero, $x-y^4$ is less than $x$, so it can't be a multiple of $x$ (whether common with $y$ or not...). Of course that holds unless you allow ZERO as a multiple, in which case $x-y^4 = 0$ would give solutions.