Can someone help me in resolving the following problem?
Find all the solutions in positive integers for the equation x-y^4 =LCM(x,y).
Thank you.
Let $\displaystyle d = \text{gcd}(x,y)$ , then $\displaystyle \text{lcm}(x,y) = \frac{xy}{d}$
Suppose there exists a positive pair of solutions x and y, note that the following must hold:
$\displaystyle x - y^4 = \frac{xy}{d} \implies dx - dy^4 = xy \implies x(d - y) = dy^4$
By assumption on positivity of x and y, $\displaystyle d - y > 0 \implies y < d$, but since $\displaystyle d|y$ (by definition), $\displaystyle d \leq y$...Contradiction!
Thus there are no positive solutions x and y to the given equation.