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I am asking about the probability that the number of Fermat primes is infinite. There is a lot of things similar to the case of Mersenne primes. But it was conjectured that the number of Mersenne primes is infinite and the number of Fermat primes is finite.

J. W. Tanner
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John2000
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  • What's the question here? – Edward Evans Jan 23 '20 at 16:25
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    Perhaps you mean "probability" instead of "possibility"? – joriki Jan 23 '20 at 16:26
  • @joriki: Yes. I mean this. – John2000 Jan 23 '20 at 16:27
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    The Fermat numbers grow very fast. So, although the factors are dramatically restricted (A prime factor of $2^{2^n}+1$ must be of the form $k\cdot 2^{n+2}+1$) , we can expect finite many Fermat primes. In fact, chances that we know all Fermat primes , are very high. The situation is completely different in the case of Mersenne numbers. The growth rate is relatively low, so we can expect infinite many Mersenne-primes. The smallest open case in the Fermat numbers is by the way $F_{33}=2^{2^{33}}+1$ – Peter Jan 23 '20 at 16:32
  • I suspect a more basic reason for the conjectures being different is that so few Fermat primes have been found. But we simply do not know whether either set is finite or infinite. – almagest Jan 23 '20 at 16:37
  • Curious : Fermat was lazy and only checked the Fermat numbers upto $F_4=65\ 537$ , which are all prime. He conjectured that all would be prime, now it seems that the opposite is true, namely that no more Fermat primes exist. – Peter Jan 23 '20 at 16:38
  • so $k$ can't be 1 mod 3 if $n$ is odd, k can't be 2 mod 3 if n is even. at least looking for prime factors. –  Jan 23 '20 at 16:40
  • @almagest Testing $F_{33}$ takes very long even with the powerful hardware we have today. Perhaps , someone finds a factor clarifying this case. Mersenne numbers can be checked far more quickly, so the next record prime will soon appear. So, the growth rate not only makes Fermat primes less likely , but also much more difficult to be proven to be prime, even if they actually are. And yes, both sets can be infinite or finite. – Peter Jan 23 '20 at 16:45
  • 3 to the power of 2 to a mersenne number you say ... –  Jan 23 '20 at 16:52
  • I think that most mathmematicians would agree that almost surely, the number of Fermat primes is finite (as we can say that Goldbach's conjecture is almost surely true although we cannot prove it). Of course, prime numbers are not actually random, but the statistical evidence is neverhteless overwhelming. Nevertheless, some mathematicians seem to conjecture that infinite many Fermat primes exist, but I have no idea how they justify this. – Peter Jan 23 '20 at 17:00
  • @Peter: Can you give an example about this: some mathematicians seem to conjecture that infinite many Fermat primes exist. – John2000 Jan 23 '20 at 17:04
  • $F_n=2M_{M_n}+3$ for comparison. –  Jan 23 '20 at 17:07
  • @Helena I cannot remember exactly the link. Maybe you google a bit around Fermat primes. – Peter Jan 23 '20 at 17:43
  • @Peter: I find only the statement in Apostol book about this. – John2000 Jan 23 '20 at 17:45
  • Euler was the first to refute Fermat by showing that $641$ divides $F_5$ – Peter Jan 23 '20 at 17:59
  • (Regarding your recent question on MathOverflow, the answer is that no one has computed the required polynomial explicitly or even, as far as the published literature indicates, worked on doing this. My apologies for the rude reception the question had there.) – Andrés E. Caicedo Jan 28 '20 at 15:09
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    @AndrésE.Caicedo: Yes. Thank you. Several mathematicians thinks that diophantine representation of number theory problems would advance our understanding of them. – John2000 Jan 28 '20 at 17:01

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