Talk:Fermat number

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It's worth noticing something. If it makes sense to talk of "probabilities" in this area, then the odds are good that Fermat's little theorem will "make" these numbers prime. That's because there is so much scope for any of the testing sequences to come up with 1 at an intermediate stage, in which case they will stick there. 1 is an attractor for the sequences. PML.

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[edit] True or false??

True or false: It is likely the F_33 will be prime with current knowledge.

Currently, the primality status of F_33 is unknown. See [1] for details. Maxal 05:45, 13 December 2006 (UTC)

[edit] 2 Squared N?

Is there a way to superscript the n above the 2 in the expression of the function? Right now it doesn't look like "Two to the two-to-the-Nth power" but "Two squared times N." That confused me greatly at first reading. Nevah 03:28, 1 May 2006 (UTC)

[edit] Factually Incorrect

From the article:

If 2n + 1 is prime, it can be shown that n must be a power of 2. (If n = ab where 1 < a, b < n and b is odd, then 2n + 1 ≡ (2a)b + 1 ≡ (−1)b + 1 ≡ 0 (mod 2a + 1).) In other words, every prime of the form 2n + 1 is a Fermat number, and such primes are called Fermat primes. The only known Fermat primes are F0,...,F4.


I don't know much about Fermat numbers but I do know that 2 is a prime number of the form 2n+1 where n is 0. Zero is not a power of two so the first part cannot be correct and the "every prime of the form 2n+1 is a fermat prime" is also incorrect since 2 is of the form 2n+1 and not a fermat prime. I'm gonna try to fix this, but somebody who knows what they're doing really should. --Shadowoftime 18:53, 16 July 2006 (UTC)

Indeed, the statement only holds for nonzero n. -- EJ 19:06, 16 July 2006 (UTC)

[edit] Regarding the phrase...

"The following heuristic argument suggests there are only finitely many Fermat primes: according to the prime number theorem, the "probability" that a number n is prime is at most A/ln(n), where A is a fixed constant. Therefore, the total expected number of Fermat primes is at most

A \sum_{n=0}^{\infty} \frac{1}{\ln F_{n}} = \frac{A}{\ln 2} \sum_{n=0}^{\infty} \frac{1}{\log_{2}(2^{2^{n}}+1)} < \frac{A}{\ln 2} \sum_{n=0}^{\infty} 2^{-n} = \frac{2A}{\ln 2}

It should be stressed that this argument is in no way a rigorous proof. For one thing, the argument assumes that Fermat numbers behave "randomly", yet we have already seen that the factors of Fermat numbers have special properties. Although it is widely believed that there are only finitely many Fermat primes, there are some experts who disagree. [2]"

Name a kind of prime that the same argument suggests is finite, but that in fact is known to be infinite. Georgia guy 20:29, 7 October 2006 (UTC)

[edit] Reciprocal sum

yo, the property "the sum of two fermat number recipricals is irrational" is clearly wrong, a rational + a rational is rational

That quote is false. The article actually says "The sum of the reciprocals of the Fermat numbers is irrational". It's the sum of all reciprocals. A sum of infinitely many rationals can be irrational. PrimeHunter 00:43, 23 December 2006 (UTC)

[edit] F33 status

2^2^33 + 1 is known to have no prime factors below... Georgia guy 23:34, 24 February 2007 (UTC)

[edit] Luca prime

A prime divisor of Fermat numbers is called Luca prime. The first two Luca prime is 3,5. It is difficult to know whether the given prime is Luca prime. It is not even known if 7 is a Luca prime. We do know that the sum of the the reciprocals of all Luca primes converges.218.133.184.93 16:27, 22 March 2007 (UTC)

The answer is that 7 is definitely not. There are certain primes that no number of the form 2^n+1 is divisible by, and 7 is such a prime. The sequence:

2, 3, 5, 9, 17, 33, 65

Is congruent to 2, 3, 5, 2, 3, 5, 2 mod 7. Georgia guy 16:57, 22 March 2007 (UTC)

I found no mention of "Luca prime" with Google. If the term has been used then it doesn't appear notable enough to mention in Wikipedia. PrimeHunter 01:16, 23 March 2007 (UTC)

Is there any algorithm to know whether the given prime is Luca prime?218.133.184.93 08:10, 23 March 2007 (UTC)

See http://primes.utm.edu/glossary/page.php?sort=FermatDivisor. Your alleged term "Luca prime" is usually called a Fermat divisor. PrimeHunter 15:28, 23 March 2007 (UTC)
The problem is that 3 and 5 are excluded from Fermat divisors.
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