# [seqfan] Re: Conjectures relating to twin primes and Lucas numbers

Farideh Firoozbakht f.firoozbakht at sci.ui.ac.ir
Fri Jan 1 22:08:33 CET 2010

```> Pleaese instead of " n = 2465 is prime, " only consider " n=2465, ".

Quoting Farideh Firoozbakht <f.firoozbakht at sci.ui.ac.ir>:

> Quoting Andrew Weimholt <andrew.weimholt at gmail.com>:
>
>> On Wed, Dec 30, 2009 at 2:01 PM, Creighton Kenneth Dement
>> <creighton.k.dement at mail.uni-oldenburg.de> wrote:
>>>
>>> I have two more variations involving Lucas numbers.
>>>
>>> Conjecture II:
>>> Let p be an odd prime.
>>> p, p+2 are twin primes if and only if
>>> p+2 divides Lucas(p+2) - 1 = A000032(p+2) - 1
>>>
>>> Conjecture III:
>>> Let n be any integer > 2.
>>> Then n, n+2 are twin primes if and only if
>>>
>>> n divides Lucas(n) - 1 = A000032(n) - 1
>>> and
>>> n+2 divides Lucas(n+2) - 1 = A000032(n+2) - 1
>>>
>>
>> For any prime p, L(p) == 1 mod p,
>> however the converse is not necessarily true.
>> L(n) == 1 mod n does not necessarily mean n is prime.
>> Composite numbers for which L(n)==1 mod n are called Lucas
>> Pseudoprimes (A005845)
>>
>> Your conjecture II is equivalent to saying
>> "if p is prime, p+2 is not a Lucas Pseudoprime"
>>
>> Your conjecture III is equivalent to saying
>> "there are no Lucas Pseudoprimes which differ by 2 from either a prime
>> or another Lucas Pseudoprime"
>>
>> Andrew
>
>
>   The smallest counterexample for Conjecture II :
>
>   p=6719 is prime, p+2|L(p+2)-1 and p+2=6721=11*13*47 isn't prime.
>
>   L(6721)-1=6721*s
>
>  s = 6004767889791519284893346140580579534741404978476561910035727177887156\
>      5001238045231945830605205631359256357619283084501345367272664451429201\
>      5316083230181540400881696622127142369952651138063373022051156286569236\
>      9290200418229655372169642183243585242258830740376370273280739775177114\
>      8868504641247335937308022553949223490665573985501930380449616439404588\
>      6025715388016549664770766108427233918279722140716981492517155966045395\
>      6312856985132437016723505839065874166043241003214599585730212965034926\
>      8337371944185116232109072473750837870982408015424448124770906618763033\
>      1602545866623685217167590424408683809370335631540717634701624102133450\
>      5626282841916075326153844796708593662069868459575639851362108414432895\
>      5300157172225419628198673709017598219081029107859970911593047987417444\
>      9584791996141143592194533294506789965334835344694737942695934564861867\
>      0154599677322915287922389638162641921758464345327760722920555114752534\
>      5735651718101710551187700187056694239112180490871110333565519684537013\
>      0668545709827245966507502187130203072514658420118538468623375483087149\
>      1868151340931049420061033159205580093678458911031979697828878147178943\
>      1117299658248121305853966517053781029488437789565700984964082860250793\
>      9649776437134398484045199757103650720037219240097032910739836632272410\
>      0655632117441747574150298949749590206192538828026365125780259687642042\
>      1994927444242992786047980735756243476857170557085861338422657936148160\
>      0
>
>
>   The smallest counterexample for Conjecture III :
>
>   n = 2465 is prime, n|L(n)-1 and n+2 is prime so n+2|L(n+2)-1
>   but obviously n, n+2 aren't twin primes.
>
>   L(2465)-1=2465*t
>
>  t = 5790503021750443850039364817554864339088813434944386276901795036999225\
>      1333118160366717065225038054607470383285122332665598517391061972889386\
>      9754201325411821191924881232862756770005478200740981883378767942406138\
>      3520612910611362775419433279171973119884210507562613140457056990457790\
>      0746084274449432736888572576851455080120388437150087165481843359462344\
>      2093748475215654145406955641034780894219006110094465770686039141623340\
>      9144969319923078976131743375116820954936767341460401659396539440690872\
>      3897221575052944528914
>
>
>   Happy new year 2010 !
>
>   Farideh
>
>

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