# Talk:Fibonacci prime

## Infinitely many

Is it the lack of spaces that is making this display incorrectly? It's missing the bottom half of the coded stuff. Mathworld gives a different reference for the GCD rule.(Michael 1964; Honsberger 1985, pp. 131-132) Divineprime

Divineprime - I now understand all of your statements about divisibility of Fibonacci numbers apart from the final sentence. This is where you say that Carmichael's theorem "does not seem to suggest that there are a finite number of examples where Fp is the one prime". I do not understand how you can use Carmichael's theorem to reach any conclusions about how many Fp (with prime index) have only one primitive factor (and so are prime) or have more than one primitive factor (and so are composite). Can you give more details of your argument ? Is this last sentence your own opinion, or do you have a reference ? Gandalf61 10:47, 15 April 2006 (UTC)

Gandalf61 - I'm glad you have a better understanding. First of all, the Fibonacci page says, "2, 3, 5, 13, 89, 233, 1597, 28657, 514229, …. It seems likely that there are infinitely many Fibonacci primes, but this has yet to be proven." Is this someone's opinion, or is there a reference?

The reference and definition of Carmichael's theorem can be thought of in explicit terms, rather than loose. "Every Fibonacci" "At least one of them" It does not state "at least x of them", where x expands at some rate. You can also look into Zsigmondy's theorem, and generalized details of the same properties. http://www.google.com/search?q=cache:h0RqXiBA72cJ:www.citebase.org/cgi-bin/fulltext%3Fformat%3Dapplication/pdf%26identifier%3Doai:arXiv.org:math/0412079+Zsigmondy+1892+&hl=en&gl=us&ct=clnk&cd=17

Divineprime - yes, it looks as if the statement that "it seems likely that there are infinitely many Fibonacci primes" on the Fibonacci number page is someone's unverified opinion. I have replaced it with "it is not known if there are infinitely many Fibonacci primes". I have also re-worded the final sentence in the Divisibility of Fibonacci numbers section to say that Carmichael's theorem does not tell us how many prime factors Fp will have, which is what I think you intended to say. And finally, you should end your contributions to Wikipedia discussion with four tildes, like this: ~~~~. This automatically signs your contributions with your user name and a timestamp, and makes discussions much easier to follow. Gandalf61 09:29, 20 April 2006 (UTC)

## 233rd Fibonacci number known to be composite??

F(7) = 13 = prime, F(13) = 233 = prime, F(233) = (if composite, what is its smallest factor??) Georgia guy 14:23, 11 July 2006 (UTC)

233 is not in the list of indices of Fibonacci primes at OEIS Sequence A001605, so F(233) is composite. Here is its factorisation, according to Ron Knott's Fibonacci pages:
F(233) = 2211236406303914545699412969744873993387956988653 = 139801 x 25047390419633 x 631484089583693149557829547141
Gandalf61 14:59, 11 July 2006 (UTC)

## Error on the page

First 2 has not been included which is both a number in the Fibonacci sequence and is a prime number. Second, 4 is included which is neither a prime number or a number in the Fibonacci sequence. Third, 7 and 11 are included which are prime numbers but are not in the Fibonacci sequence. —Preceding unsigned comment added by Agnostic 4 Now (talkcontribs)

I guess you refer to the text:

"It is not known if there are infinitely many Fibonacci primes. The first 33 are Fn for the n values (sequence A001605 in the OEIS):

3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839."

Note that the list is not the Fibonacci primes Fn but their indices n. The corresponding Fibonacci primes Fn have up to 17103 digits for n=81839 so it would be impractical to list their decimal expansions. The page is correct. PrimeHunter (talk) 23:29, 15 May 2009 (UTC)

## All Fibonacci series

Are all series considered?

0,2,2,4,6,10,26,36,62

0,3,3,6,9,15,24,39,63

divergence from convergence may be more interesting. Mydogtrouble (talk) 21:25, 15 October 2009 (UTC)

## Known Fibonacci Prime Section Flawed

The section listing known Fibonacci primes is erroneous and misleading. It seems to be paraphrased from the following:

"The first few proven prime Fibonacci numbers F_n are 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... (Sloane's A005478), which occur for n=3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839, 104911, 130021, 148091, 201107, 397379, ... (Sloane's A001605; Dubner and Keller 1999), where the Fibonacci numbers with indices 104911, 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721, ... are probable primes (Caldwell)." [1]

Obviously 4 is neither a prime nor a Fibonacci number. It seems that the author was perhaps confused by the source. I do not know enough about the subject to correct it without leaving other errors but I want to point it out. I wish I could help more.--Rotellam1 (talk) 16:04, 9 September 2013 (UTC)

Please notice this comment from the source:
"Since F[n] divides F[mn] (cf. A001578, A086597), all terms of this sequence are primes except for a(2)=4 (=2*2 but F[2]=1). - M. F. Hasler, Dec 12 2007"
So the 4th term is a special case. — Glenn L (talk) 17:22, 9 September 2013 (UTC)
"The first 33 are Fn for the n values (sequence A001605 in the OEIS):
3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839."
It seems very clear to me that "Fn for the n values" means that the Fibonacci primes are F3, F4, F5, F7, F11, F13, F17, F23, F29, F43, F47, F83, F131, F137, F359, F431, F433, F449, F509, F569, F571, F2971, F4723, F5387, F9311, F9677, F14431, F25561, F30757, F35999, F37511, F50833, F81839
I think the list would be harder to read like that but several people have made your false "correction" of removing 4, so right after 4 I once added [1] the source comment "This prime is F_4 = 3 and should not be removed". This reduced (but didn't completely stop [2]) the false corrections, but readers don't see the comment. For each reader who makes the false correction or posts it to the talk page, there are probably many others who think we are wrong but don't do anything. The article already says: "Except for the case n = 4, all Fibonacci primes have a prime index". Do we really need to explain further that when we say "Fn for the n values", n is not the Fibonacci number but its index, and the index n = 4 gives the Fibonacci number F4 = 3, and 3 is prime although 4 is not. PrimeHunter (talk) 23:18, 9 September 2013 (UTC)

References

1. ^ Weisstein, Eric W. ""Fibonacci Prime"". MathWorld--A Wolfram Web Resource. Retrieved 9 Sept. 2013. Check date values in: |access-date= (help)

## Why?

"(This implies the infinitude of primes.)" ????? — Preceding unsigned comment added by 186.140.103.166 (talk) 16:59, 15 September 2013 (UTC)

If there were only a finite number n of primes p then the n values Fp would have distinct prime factors, so each Fp would have to be one of the n primes. This is not the case. PrimeHunter (talk) 23:47, 15 September 2013 (UTC)
PrimeHunter, This does not make sense. You will have to be clearer otherwise this section should be deleted.
It has very little to do with the subject, unless you can verify your original research. Primedivine (talk) 01:10, 27 August 2015 (UTC)
186.140.103.166 was asking why Fibonacci prime#Divisibility of Fibonacci numbers says "(This implies the infinitude of primes.)" It's not about the conjectured infinitude of Fibonacci primes but about the well-known infinitude of all primes. My answer is trivial but Wikipedia:Original research does not apply to talk pages so I don't have to dig up a source for this trivial observation. PrimeHunter (talk) 01:41, 27 August 2015 (UTC)
Well, of course you were talking about the infinitude of primes, not Fibonacci primes. Why would you think that? You are wrong if you think that.
Why does it makes sense to have this in this article anyway?
You can say what you want on a talk page, but it's still not related to the subject enough without some kind of reference. It needs one, or else.
I am curious why you would not want to explain it, if it is so trivial. If a person asked you for the time of day, would you grumble off and say "the rules state that I don't have tell you what time it is, because the time of day is trivial." You have already misunderstood me as thinking the statement meant Fibonacci primes, and you were wrong about that. — Preceding unsigned comment added by Primedivine (talkcontribs) 03:58, 27 August 2015 (UTC)

## Proposed update to Divisibility of Fibonacci numbers

If and only if a prime p congruent to 1 or 4 (mod 5), then p divides Fp-1, otherwise, p divides Fp+1. (The only exception is p = 5, if and only if p = 5, then p divides Fp)

Fibonacci numbers that have a prime index p do not share any common divisors greater than 1 with the preceding Fibonacci numbers, due to the identity

GCD(Fn, Fm) = FGCD(n,m).[1]

(This implies the infinitude of primes.)

For n ≥ 3, Fn divides Fm iff n divides m.[2]

If we suppose that m is a prime number p from the identity above, and n is less than p, then it is clear that Fp, cannot share any common divisors with the preceding Fibonacci numbers.

GCD(Fp, Fn) = FGCD(p,n) = F1 = 1

This means that Fp will always have characteristic factors or be a prime characteristic factor itself. The number of distinct prime factors of each Fibonacci number can be put into simple terms.

• 1. "Fnk is a multiple of Fk for all values of n and k from 1 up."[3]
It's safe to say that Fnk will have "at least" the same number of distinct prime factors as Fk.
All Fp will have no factors of Fk, but "at least" one new characteristic prime from Carmichael's theorem.
• 2. Carmichael's Theorem applies to all Fibonacci numbers except 4 special cases. {except for 1, 8 and 144}

"If we look at the prime factors of a Fibonacci number, there will be at least one of them that has never before appeared as a factor in any earlier Fibonacci number."

Let πn be the number of distinct prime factors of Fn.[4]
If k | n then πn >= πk+1. {except for π6 = π3 = 1}
If k=1, and n is an odd prime, then 1 | p and πp >= π1+1, or simply put πp>=1.
n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Fn 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657
πn 0 0 0 1 1 1 1 1 2 2 2 1 2 1 2 3 3 1 3 2 4 3 2 1

The first step in finding the characteristic quotient of any Fn is to divide out the prime factors of all earlier Fibonacci numbers Fk for which k | n.[5]

The exact quotients left over are prime factors that have not yet appeared.

If p and q are both primes, then all factors of Fpq are characteristic, except for those of Fp and Fq.

GCD(Fpq, Fq) = FGCD(pq,q) = Fq
GCD(Fpq, Fp) = FGCD(pq,p) = Fp
πpq>=πqp+1 {except for πp2>=πp+1}

For example, F247 π(19*13)>=(π1319)+1.

The number of distinct prime factors of the Fibonacci numbers with a prime index is directly relevant to the counting function.()

p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
πp 0 1 1 1 1 1 1 2 1 1 2 3 2 1 1 2 2 2 3 2 2 2 1 2 4

Primedivine (talk) 13:51, 5 September 2015 (UTC)

Is there any known reference to the recursive counting function?

Let d be the divisors of n, except 6 and 12 since these Fibonacci indexes don't produce any new characteristic factors.
Sum the number distinct prime factors of each F(d).
For each composite divisor d except 4, subtract the number of distinct prime factors for each Fibonacci number with an index that is a prime divisor of the divisor d, or 4.
However, if the divisor of d is composite and not 4, then repeat this process until the factor of the divisor is a prime number, or 4.
For example n=315, and d=3,5,7,9,15,21,35,45,63,105. The total for n=315 is 31.
This is a simple recurring expression. Note: {} composite, [] prime.
• [π(3)] = 1
• [π(5)] = 1
• [π(7)] = 1
• {π(9) - [π(3)]} = 1
• {π(15) - [π(3)] - [π(5)]} = 1
• {π(21) - [π(3)] - [π(7)]} = 1
• {π(35) - [π(5)] - [π(7)]} = 1
A slightly more complex recurrsion that nests the same function when encountering composite divisors, ie:
• {π(45) - [π(3)] - [π(5)] - {π(9)-[π(3)]} - {π(15)-[π(3)]-[π(5)]} } = 1
• {π(63) - [π(3)] - [π(7)] - {π(9)-[π(3)]} - {π(21)-[π(3)]-[π(7)]} } = 1
• {π(105) - [π(3)] - [π(5)] - [π(7)] - {π(15)-[π(3)]-[π(5)]} - {π(21)-[π(3)]-[π(7)]} - {π(35)-[π(5)]-[π(7)]} } = 1
This accounts for all characteristic factors and exceptions collected along the way, from earlier Fibonacci numbers.

Primedivine (talk) 18:27, 15 September 2015 (UTC)

References

1. ^ Paulo Ribenboim, My Numbers, My Friends, Springer-Verlag 2000
2. ^ Wells 1986, p.65
3. ^ The mathematical magic of Fibonacci numbers Factors of Fibonacci numbers
4. ^ Online encyclopedia of integer sequences The number of distinct prime factors of the n-th Fibonacci number
5. ^ Jarden - Recurring sequences, Volume 1, Fibonacci quarterly, by Brother U. Alfred

## Reciprocal fibonacci primes

This was undone [3] because of formal reason, but the fact is trivial. Adding the first relevant fibonacci primes (→Official OEIS list) – you can do it in Excel within one and a half minute. The result converges extreme fast to a constant number: ${\displaystyle 1.12644722767285...}$. — Preceding unsigned comment added by 176.5.31.69 (talkcontribs) 19:49, 21 October 2016 (UTC)

Whether this sum is convergent depends on the behavior of the entire series (if there are infinitely many), not on the first few terms. The reciprocal of all primes diverges, so the result is not trivial at least. Of course if the pattern of large ratios between subsequent terms continues, it will converge, but adding these things to the article requires a reliable source. Gap9551 (talk) 20:08, 21 October 2016 (UTC)
The sum must be less than the Reciprocal Fibonacci constant so it actually is trivial that it converges. But if reliable sources haven't bothered to mention it then I don't think we should either. The sum of the reciprocals of the indices of Fibonacci primes would be more interesting. I don't know whether convergence has been proved. PrimeHunter (talk) 21:04, 21 October 2016 (UTC)
Good point, thanks. Gap9551 (talk) 17:32, 22 October 2016 (UTC)

## Primitive part definition conflict

There is a conflict in the definition and references. The OEIS refers to a pre-print by Pomerance/Wagner, https://math.dartmouth.edu/~carlp/fibinttalk.pdf (page 12-13),where the primitive part is always the product of primitive prime factors. This product dividing the symbol ${\displaystyle \phi _{n}}$. Most of the time ${\displaystyle \phi _{n}}$ is the primitive part depending on the form of n. This agrees with Chris Caldwell's definition, where he says the symbol ${\displaystyle \phi _{n}}$ is the primitive part(or Primitive factor, ie not the primitive part). However OEIS defines the primitive part as always the symbol ${\displaystyle \phi _{n}}$. Clearly this is wrong, otherwise the equation on page 13 doesn't make sense, ie ${\displaystyle \phi _{p^{k}\alpha (p)}}$= ${\displaystyle p}$ x (the primitive part of ${\displaystyle F_{p^{k}\alpha (p)}}$).