Talk:Fermat number

Proof of the theorem of Edouard Lucas

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)

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 a^2k+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)

Excuse me, I mean : from the last formula I get (1+2^2n-1)². 2^{- 2n-1}=2 (mod p), then 2 is a square if n > 1 (without using the fact that odd powers of 2 are residues mod p, I use only that 2 is invertible mod p). JFM —Preceding unsigned comment added by 79.93.171.86 (talk) 17:24, 22 December 2010 (UTC)

In fact, we can get an explicit value of r such that r^2 = 2 (mod Fn). Let i = 2^(2^(n-1)). Then i^2 = -1 (mod Fn). i exists for n≥1. Let q = 2^(2^(n-2)). Then q^2 = i, and q^4 = -1 (mod Fn). q exists for n≥2. Now (i-1)^2 = -2i (mod Fn) by direct algebraic expansion. So let r = q(i-1). Then r^2 = 2 (mod Fn). r exists for n≥2. 86.4.253.180 (talk) 00:45, 20 September 2013 (UTC)

In its present form this sketch of the proof is not well organized and has a significant gap. One needs precisely that the multiplicative order of 2 in G is 2ⁿ⁺¹, not merely that this is divisible by the order of 2, to conclude p-1 is divisible by that power of 2. If no one objects, I will fix that gap. Further it seems a bit misplaced to lump this proof sketch with "other results" when it has been mentioned more than once already, including early in the article as a whole. Hardmath (talk) 14:27, 23 April 2019 (UTC)

Perhaps a simpler way to get an explicit value of r such that r^2 = 2 (mod Fn) is to define q = 2^(n-2), which clearly exists for n≥2. Then 4q = -1 (mod Fn), and let r = 2^(3q) - 2^q (mod Fn). Then by direct algebra: r^2 = 2^(6q) - 2*2^(4q) + 2^(2q) = (2^(4q) + 1)* 2^(2q) - 2*2^(4q)= Fn * 2^(2q) - 2*(Fn - 1) = 2 (mod Fn). 86.11.96.95 (talk) 01:36, 11 February 2022 (UTC)

Non-sensical heuristic for the infinitude of Fermat primes

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)

Sequence of Fermat numbers

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)

Main motivation of study by Fermat

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)

I understand that main aspect needing a citation concerns the simple enumeration procedure used by Fermat when formulating the conjecture.--109.166.129.57 (talk) 15:13, 10 September 2019 (UTC)

F5 factorization date

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 ([[1]]). 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)

It is proven that there are only 5 Fermat primes.

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)

Please read the article.—Anita5192 (talk) 20:23, 8 October 2021 (UTC)

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)

Bad definition

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)

Semi-protected edit request on 12 November 2021

Vizz01 (talk) 05:05, 12 November 2021 (UTC)

Request denied. You did not specify a "complete and specific description." The next time you make a request, please read the instructions that were in the template above.—Anita5192 (talk) 05:58, 12 November 2021 (UTC)

I added F12 entry (last line in table below) at https://wiki.riteme.site/wiki/Fermat_number#Factorization_of_Fermat_numbers

What can be more "complete and specific description" ? Anita5192 Can you please elaborate ? Here is the edit again since it most likely disappeared when the request was denied.

F0	=	21	+	1	=	3 is prime
F1	=	22	+	1	=	5 is prime
F2	=	24	+	1	=	17 is prime
F3	=	28	+	1	=	257 is prime
F4	=	216	+	1	=	65,537 is the largest known Fermat prime
F5	=	232	+	1	=	4,294,967,297
					=	641 × 6,700,417 (fully factored 1732 )
F6	=	264	+	1	=	18,446,744,073,709,551,617 (20 digits)
					=	274,177 × 67,280,421,310,721 (14 digits) (fully factored 1855)
F7	=	2128	+	1	=	340,282,366,920,938,463,463,374,607,431,768,211,457 (39 digits)
					=	59,649,589,127,497,217 (17 digits) × 5,704,689,200,685,129,054,721 (22 digits) (fully factored 1970)
F8	=	2256	+	1	=	115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129, 639,937 (78 digits)
					=	1,238,926,361,552,897 (16 digits) × 93,461,639,715,357,977,769,163,558,199,606,896,584,051,237,541,638,188,580,280,321 (62 digits) (fully factored 1980)
F9	=	2512	+	1	=	13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,0 30,073,546,976,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,6 49,006,084,097 (155 digits)
					=	2,424,833 × 7,455,602,825,647,884,208,337,395,736,200,454,918,783,366,342,657 (49 digits) × 741,640,062,627,530,801,524,787,141,901,937,474,059,940,781,097,519,023,905,821,316,144,415,759, 504,705,008,092,818,711,693,940,737 (99 digits) (fully factored 1990)
F10	=	21024	+	1	=	179,769,313,486,231,590,772,930...304,835,356,329,624,224,137,217 (309 digits)
					=	45,592,577 × 6,487,031,809 × 4,659,775,785,220,018,543,264,560,743,076,778,192,897 (40 digits) × 130,439,874,405,488,189,727,484...806,217,820,753,127,014,424,577 (252 digits) (fully factored 1995)
F11	=	22048	+	1	=	32,317,006,071,311,007,300,714,8...193,555,853,611,059,596,230,657 (617 digits)
					=	319,489 × 974,849 × 167,988,556,341,760,475,137 (21 digits) × 3,560,841,906,445,833,920,513 (22 digits) × 173,462,447,179,147,555,430,258...491,382,441,723,306,598,834,177 (564 digits) (fully factored 1988)
F12	=	24096	+	1	=	1,044,388,881,413,152,506,691,75...243,804,708,340,403,154,190,337 (1234 digits)
					=	114,689 × 26,017,793 × 63,766,529 × 190,274,191,361 × 1,256,132,134,125,569 × 568,630,647,535,356,955,169,033,410,940,867,804,839,360,742,060,818,433 × 40,386,086,203,521,847,842,442,0...117,418,021,434,702,300,430,337 (1133 digits) (last factor is a not-yet-factored composite)

 Not done: @Vizz01: The 1213-digit factor is composite. It has the known prime factors 190274191361 and 1256132134125569, but it's still not fully factored. Your first request [2] was rejected because it was empty. PrimeHunter (talk) 07:46, 13 November 2021 (UTC)

Agreed and updated the most complete known factors above. Thanks PrimeHunter for your very valid point.

@Vizz01: We list F₀ to F₁₁ and say "only F₀ to F₁₁ have been completely factored. I think it would be a little odd to stop at F₁₂. All numbers from F₁₂ to F₁₉ are partially factored. We link http://www.prothsearch.com/fermat.html which has all known factors. PrimeHunter (talk) 06:52, 16 November 2021 (UTC)

References

Header intro-section, missing reference/link

In the header/intro section we have: “If 2^k + 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)

Finite fermat primes, but irrational reciprocal sum?

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)

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)