RO-TST 2010 Problem 7

(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 $n > 2$ , 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 $k$ .

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