Talk:Fermat number

message: In the sketch of proof of the theorem I propose to replace the last sentence :
"Since an odd power of 2 is a quadratic residue (mod p), so is 2 itself."
by 
"Since an even power of 2 is a quadratic residue (mod p), so is 2 itself."
which seems (to me) more accurate. JFM79.93.171.237 (talk) 15:30, 16 December 2010 (UTC)
replies: text: No, that's an incorrect argument. An even power of every element is a quadratic residue, this does not imply anything about the element's being quadratic residue. If an odd power a2k+1 = b2 is a quadratic residue, then a is a quadratic residue, since a = (ba−k)2. Anyway, the power in question, 1 + 2n−1, is odd indeed.—Emil J. 15:45, 16 December 2010 (UTC)
thread_title:  Proof of the theorem of Edouard Lucas  

message: The heuristic for the infinitude starts of with : "Suppose we regard the conditional probability that n is prime, given that we know all its prime factors are 1 modulo M, as at least CM/ln(n)."
This is non-sense. The number of integers up to x all of whose prime factors are congruent to 1 mod M is equal to roughly x / (\log x)^{1 - 1/\varphi(M)} on the other hand the number of integer that are prime and all of whose prime factors are congruent to 1 mod M is (1/\varphi(M)) * x / (\log x). Therefore the conditional probability of x being prime given that all of its prime factors are = 1 (mod M) is the latter divided by the former, that is
(\log x)^{-1/\varphi(M)} / \varphi(M). Taking x = F_n and M = 2^n we now see that this conditional probability summed over all n converges very nicely. 
I am therefore removing the non-sensical heuristic for the infinitude.  — Preceding unsigned comment added by 132.204.90.25 (talk) 03:54, 5 March 2019 (UTC)
thread_title: Non-sensical heuristic for the infinitude of Fermat primes

message: I notice editor DVdm has reverted the specification that these numbers are terms in a numerical sequence as non-beeing sourced. I think that this aspect is easily noticeable without source like the statement "The cloudless sky is blue" which is considered a common fact not needing citation, as specified in some wikirules.--109.166.129.57 (talk) 14:32, 10 September 2019 (UTC)
thread_title:  Sequence of Fermat numbers 

message: Among the edits reverted by user DVdm is the specification of main motivation of the study of these type of numbers by Fermat which I agree it might need a citation from a source. The info inserted in the introductory paragraphs of the article is already in a section below, but without explicit acknowledgement of main motivation of study.--109.166.129.57 (talk) 14:38, 10 September 2019 (UTC)
thread_title:  Main motivation of study by Fermat 

message: I've asked for clarification about the date of 1732 for the "full factorizaton" of $F_5$.  Although Euler found the factor 641 in 1732, there is no indication he even tried to prove the other factor 6700741 prime. Refer to C. Edward Sandifer's book How Euler Did Even More ([]). The earliest I've heard this number being published as prime is with Dase's catalog in 1861.— Preceding unsigned comment added by Casu Marzu (talk • contribs) 16:55, 13 April 2020 (UTC)
thread_title:  F5 factorization date 

message: See this article, Fermat numbers 2^(2^n)+1 is not prime for n>=5, thus there are only 5 Fermat primes: 3, 5, 17, 257, 65537. 2402:7500:92C:4EB9:78DA:44AC:3879:8299 (talk) 19:22, 8 October 2021 (UTC)
replies: text: Please read the article.—Anita5192 (talk) 20:23, 8 October 2021 (UTC)
replies: text: There are infinity solution in Real numbers pair k1 & k2, but no guarantee in Natural numbers where are we looking for --Ming mm (talk) 13:58, 14 October 2021 (UTC)
thread_title:  It is proven that there are only 5 Fermat primes. 

message: I do believe that Fermat numbers are defined as $2^n + 1$, and NOT as $2^{2^n} + 1$.  Fermat primes are primes that happen to be Fermat numbers... and they (proven) take the form $2^{2^m}+1$.  Thus Fermat primes are prime Fermat numbers of the form $2^n+1$ where necessarily $n=2^m$.  — Preceding unsigned comment added by 134.204.1.226 (talk) 20:22, 5 November 2021 (UTC)
thread_title:  Bad definition 

message: In the header/intro section we have:
“If 2k + 1 is prime and k > 0, then k must be a power of 2”
which is not justified or referenced. It would be useful to link to the section below:
Other theorems about Fermat numbers
where the introductory statement is immediately proved.
91.125.173.22 (talk) 11:51, 13 November 2021 (UTC)
thread_title:  Header intro-section, missing reference/link 

message: The page states at the same time that heuristics suggest F4 to be the last Fermat prime and that the sum of the reciprocals of the Fermat primes is irrational. This can't happen if there are finitely many of them. 69.176.153.154 (talk) 00:23, 11 January 2023 (UTC)
replies: text: It does not specify that the sum must include only primes. All the composite Fermat numbers (e.g. 4294967297) are includeed as well –LaundryPizza03 (dc̄) 00:25, 11 January 2023 (UTC)
thread_title:  Finite fermat primes, but irrational reciprocal sum?