Question 76
(a) Given a positive integer $k$, prove that there do not exist two distinct integers in the open interval $(k^2, (k+1)^2)$ whose product is a perfect square.
(b) Given an integer $n \gt 2$, prove that there exist $n$ distinct integers in the open interval $(k^n, (k+1)^n)$ whose product is the $n$-th power of an integer, for all but a finite number of positive integers $k$.
(b) Given an integer $n \gt 2$, prove that there exist $n$ distinct integers in the open interval $(k^n, (k+1)^n)$ whose product is the $n$-th power of an integer, for all but a finite number of positive integers $k$.