Does there exist infinitely many positive integers \(n\) such that the first digit of \(2^n\) is \(9\)?

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Does there exist infinitely many positive integers \(n\) such that the first digit of \(2^n\) is \(9\)?

Prove that for all positive integers m and n, there is a positive integer k such that \[ (\sqrt{m}+\sqrt{m-1})^n = \sqrt{k}+\sqrt{k-1}.\]

Prove that each positive integer x can be written as a sum of at most 53 integers to the fourth power. In other words, for every positive integer x, there exist \(y_1,y_2,\ldots,y_k\) with \(k\le 53\) such that \(x=\sum_{i=1}^k y_i^4\).