Competition problem — Number theory

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Number theory Difficulty 6.1 National olympiad Prove it

17. Let $n \geqslant 0, F_{n}=2^{2^{n}}+1$ (it is called a Fermat number); and let $m \neq n$ . Prove: if $d>1$ , and $d \mid F_{n}$ , then $d \nmid F_{m}$ . From this, deduce that there are infinitely many primes.

Solution

17. When $m>n$ , it must be that $F_{n} \mid 2^{2^{m}}-1$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.