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

Farideh Firoozbakht f.firoozbakht at sci.ui.ac.ir
Fri Jan 1 22:02:39 CET 2010

```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

isn't correct, because:

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