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November 27

Fermat primes possibly infinite??

Read the Fermat number article. It has 2 arguments, both of which suggest that there are finitely many Fermat primes. For the first argument, it says that it isn't a proof simply because Fermat number factors have special properties. But it says nothing about the second argument not being a proof. Can anyone add to the article the reason it's not a proof?? Georgia guy (talk) 15:36, 27 November 2019 (UTC)[reply]

I assume you're referring to the section "Heuristic arguments for density". Really the only difference between the two arguments is that the second one allows for the fact that we have a lower bound on the size of the factors. This may give a better estimate for the "expected" number of primes, but it's still based on the shaky assumption that whether the number is prime or not is a random occurrence. There are a number of theorems that describe the statistics of the distribution of primes, e.g. the Prime number theorem and Dirichlet's theorem on arithmetic progressions, and that allows you to make predictions on the behavior of random sequences that have the same statistical properties. But the actual primes are not random so a conclusion about random sequences does not necessarily apply to them. --RDBury (talk) 08:47, 28 November 2019 (UTC)[reply]

Those are heuristics based on the Cramér random model, which empirically describe the approximate distribution of primes, but the exact distribution is very mysterious. A rigorous proof about the number of Fermat primes would likely be extremely technical and complicated. 67.164.113.165 (talk) 10:50, 28 November 2019 (UTC)[reply]