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Suppose that the Fermat numbers $F_m$ are pairwise relatively prime.

Can someone help me prove, given this, that there are infinitely many primes.

jento1
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  • There is a question at https://math.stackexchange.com/questions/1642835/how-does-the-fact-that-fermat-primes-are-relatively-prime-imply-there-are-infini that might have an answer that answers your question. I will answer it here but I suspect everybody's asking rate is limited. If you ask questions that you can easily find the answer to, it may be one less of another question you have that you will get a chance to ask if you keep having more questions than you're able to ask in a given amount of time. I've sometimes found answers to Stack Exchange questions with a Google search. Also, I've – Timothy Mar 18 '20 at 02:31
  • sometimes had a question and then found another Stack Exchange question through the review I go through when I'm about to ask a question of other questions similar to it that might answer by question and then found that I got the answer I was looking for and so didn't ask it after all. Also, some users probably spend more time answering questions that weren't already answered in an answer of another question and spend very little time to answer questions like this one. Never mind, I think Brian's answer already did an excellent job of answering your question. If that still doesn't solve your – Timothy Mar 18 '20 at 02:36
  • problem, there's nothing I can do. On Earth Science Stack Exchange, somebody was discouraged from answering my questions because they saw that I hadn't put a check mark on the answers to any of my questions. Somebody might do the same for your questions. If you put a check mark beside an answer that didn't solve your problem, they might start writing answers to other questions of yours that don't solve your problem. – Timothy Mar 18 '20 at 02:41

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Since the Fermat numbers are relative prime, they all have distinct prime factors. Since there are infinitely many Fermat numbers, this means that there are necessarily infinitely many prime numbers.

Brian
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