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Home » Olympiad Section » Number Theory » Number Theory Open Questions
 
Fermat-Lucas-Carmichaels
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wpolly
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#1
Fermat-Lucas-Carmichaels
research
A positive integer is called Fermat-Lucas-Carmichael Number iff:
1) it's odd, squarefree, and have at least three distinct prime divisors;
2) for every prime p|n, we have p+1|n+1 and p-1|n-1.

construct an example of Fermat-Lucas-Carmichael number or give a proof that such numbers don't exist.

Note: those numbers are both Fermat-pseudoprime to any base and Lucas-pseudoprime to any Lucas Sequence.

PostPosted: Tue Nov 16, 2004 3:28 am  Back to top 
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RobertuX
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#2
Re: Fermat-Lucas-Carmichaels
research
wpolly wrote:
A positive integer is called Fermat-Lucas-Carmichael Number iff:
1) it's odd, squarefree, and have at least three distinct prime divisors;
2) for every prime p|n, we have p+1|n+1 and p-1|n-1.

construct an example of Fermat-Lucas-Carmichael number or give a proof that such numbers don't exist.

Note: those numbers are both Fermat-pseudoprime to any base and Lucas-pseudoprime to any Lucas Sequence.

Of course, is an interesting problem, is really to probe that if exists an exmaple of course must veriify

0. n odd,
1. \mu(n)\ne 0, Moebious function,
2. n=p(1+\fbox{p-1}(1+p)), n+1=(1+\fbox{p-1}p)(p+1) and n-1=(1+\fbox{p+1}p)(p-1), where \fbox{p+1} and \fbox{p-1} are the no commun factors o p+1 and p-1 respectively.


Good luck to find some example.

B.R.
RX TCM
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RobertuX against the PEOPLE WHO try to make other people to HATES Science

PostPosted: Wed Feb 16, 2005 4:36 am  Back to top 
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rgep
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#3
This is discussed in "Unsolved Problems in Number Theory" (3rd edition) by Richard Guy (Springer Verlag, 2004) in section A13. I believe it is still an open problem.

PostPosted: Tue Aug 23, 2005 12:43 pm  Back to top 
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