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Topic: Lehmer's problem
Replies: 35   Last Post: Oct 13, 2001 9:25 AM

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 Jan Kristian Haugland Posts: 1,303 Registered: 12/4/04
Re: Lehmer's problem
Posted: Oct 5, 2001 9:00 AM

Phil Carmody wrote:
>
> Jan Kristian Haugland wrote:

> > > > http://home.hia.no/~jkhaug00/carmichael.txt
>
> > Hopefully it's up already. The server seems to have
> > frequent but short breakdowns. Two highlights:
> >
> > n1 = 6840001 = 7 x 17 x 229 x 251
> >
> > gives (n1-1)/phi(n1) = 5/4, and
> >
> > n2 = 55462177 = 17 x 23 x 83 x 1709
> >
> > gives (n2-1)/phi(n2) = 9/8.
> >
> > Non-Carmichael numbers may also be worth looking
> > at: For instance, 3200 / phi(3201) = 5/3.

>
> Cool.
>
> Using your notation, "c1 c2 r1/r2 p1 x p2 x p3", is the following
> true?
>
> c1 is a carmichael iff gcd(r1,c1)=1 ?

All the entries on the web page are Carmichael numbers.
The 3201 comes from a little search I did on the side
earlier today. c2 is phi(c1) and r1/r2 is the reduced
form of (c1-1)/c2. Since r1 divides c1-1, the value of
gcd(r1, c1) is necessarily 1. p1 x p2 x ... is the
factorization of c1.

--

J K Haugland
http://home.hia.no/~jkhaug00