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Consider the Fermat numbers ##F_n=2^{2^n}+1##.
- Prove: $$\prod_{k=0}^{n-1}F_k = F_n -2\quad (n\geq 1)$$
- Conclusion:
If ##m\,|\,F_k\;\;(k<n)## and ##m\,|\,F_n## then ##m\,|\,2##. Since ##F_n## are odd, ##m=1##. Hence ##F_k## and ##F_n## are coprime.
Which famous result follows immediately as corollary?