[seqfan] Re: More (composite) terms for A233281. Was: Is Fibo(p) always squarefree?

[Antti Karttunen]

On Sun, Feb 2, 2014 at 4:01 AM, Antti Karttunen
<antti.karttunen at gmail.com> wrote:
> Excuse me,
>
> but seeing that the topic was at least tangentially about A001177
> (Fibonacci entry points), I wonder, is there anybody who (with his/her
> desktop super-computer) could search for more composite terms in:
> http://oeis.org/A233281 that I recently submitted: "Numbers n such
> that A001177(n) is prime."
> So far, only two composites there, 4181 and 10877, are known.
>
> Also, another idea: If we collect a subset from
> http://oeis.org/A061488 the primitive prime factors, but only for the
> composite Fibonacci numbers, we should get:
> 3,7,17,11,29,61,...
> (Not yet in OEIS, but warning: my hand/head-calculation at 3:50 am local time).

Actually, use:
http://oeis.org/A061446
E.g. A061446(19) = 4181 = 37*113, both should be included at that
point, not just 37 as in A061448.


> "Primitive prime divisors of fibonacci(c) with c composite, ordered by c".
> which I think should be a complementary subset (among primes)
> to http://oeis.org/A092395 "Primes occurring as divisors of
> fibonacci(p) with p prime." ?
>
>
> Yours,
>
> Antti Karttunen
>
>
>>
>> Message: 7
>> Date: Tue, 28 Jan 2014 14:40:49 -0500
>> From: Charles Greathouse <charles.greathouse at case.edu>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: Is F(p) always squarefree?
>> Message-ID:
>>         <CAAkfSGLwO4e434QS8fwqA0SL4M8n1aeOtBAZCyhre8Ed6bZLDg at mail.gmail.com>
>> Content-Type: text/plain; charset=ISO-8859-1
>>
>> Suppose F_n is divisible by k^2. Then n is divisible by A001177(k^2) =
>> A132632(k). So a necessary condition for F_p being squarefree is that
>> A132632(q) is prime for some prime q. But this can happen only when Wall's
>> conjecture fails, so if F_p is not squarefree than it is divisible by the
>> square of a Wall-Sun-Sun prime. (Right?) I think current expectations are
>> that infinitely many Wall-Sun-Sun primes exist, but they should have only
>> doubly-logarithmic density and so it seems very hard to find any and
>> near-impossible to find more than one.
>>
>> Charles Greathouse
>> Analyst/Programmer
>> Case Western Reserve University
>>
>>
>> On Tue, Jan 28, 2014 at 2:17 PM, Alonso Del Arte
>> <alonso.delarte at gmail.com>wrote:
>>
>>> Given a prime p, the number Fibonacci(p) might be composite, but, at least
>>> for small p, appears to always be squarefree. This seems like something
>>> that could easily be proven one way or the other with something in Koshy's
>>> book, but the Library is closed today.
>>>
>>> Al
>>>
>>> --
>>> Alonso del Arte
>>> Author at SmashWords.com<
>>> https://www.smashwords.com/profile/view/AlonsoDelarte>
>>> Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>
>>
>> ------------------------------
>>
>> Message: 8
>> Date: Tue, 28 Jan 2014 12:12:44 -0800
>> From: "T. D. Noe" <noe at sspectra.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: Is F(p) always squarefree?
>> Message-ID: <p06240808cf0dbf0b404a@[192.168.1.2]>
>> Content-Type: text/plain; charset="us-ascii"
>>
>> There is a paper "The Fibonacci sequence module p^2 - An investigation by
>> computer for p < 10^14" by Elsenhans and Jahnel.  See
>> http://www.uni-math.gwdg.de/tschinkel/gauss/Fibon.pdf
>>
>> Best regards,
>>
>> Tony
>>
>> At 2:17 PM -0500 1/28/14, Alonso Del Arte wrote:
>>>Given a prime p, the number Fibonacci(p) might be composite, but, at least
>>>for small p, appears to always be squarefree. This seems like something
>>>that could easily be proven one way or the other with something in Koshy's
>>>book, but the Library is closed today.
>>>
>>>Al
>>>
>>>--
>>>Alonso del Arte
>>>Author at
>>>SmashWords.com<https://www.smashwords.com/profile/view/AlonsoDelarte>
>>>Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>>>
>>>_______________________________________________
>>>
>>>Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>>