2011 USAMO 第 4 题

Consider the assertion that for each positive integer $n \ge 2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of 4. Either prove the assertion or find (with proof) a counter-example.

考虑这样的断言：对于每个正整数 $n \ge 2$，$2^{2^n}$ 除以 $2^n-1$ 的余数是 4 的幂。要么证明该断言，要么找到(带证明)反例。