Problem 3, 2012

For every natural number \(n\), let \(x_n\) be the number obtained by writing the squares of the first \(n\) natural numbers one after another, in increasing order; for example

Call a natural number \(y\) a perfect power if \(y = a^k\) for some natural numbers \(a\) and \(k > 1\). Prove that there exist infinitely many natural numbers \(n\) for which \(x_n\) is not a perfect power.