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Topic: PNT look-alikes
Replies: 15   Last Post: May 19, 2011 2:45 AM

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lwalke3@lausd.net

Posts: 2,394
Registered: 8/3/07
Re: PNT look-alikes
Posted: May 18, 2011 2:24 PM
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On May 18, 11:16 am, Transfer Principle <lwal...@lausd.net> wrote:
> By Fermat's Little Theorem (Carmichael generalization), the number
> of digits after which 1/m repeats must divide lambda(m), which in
> turn divides phi(m). It follows that if m is "decimal-maximal" (or
> "base-b-maximal to any base b), it must be _prime_.
> http://en.wikipedia.org/wiki/Repeating_decimal
> http://en.wikipedia.org/wiki/Carmichael_function


OK, I found a few more relevant Wikipedia pages:

First, the elements of M are called "full reptend primes":

http://en.wikipedia.org/wiki/Full_reptend_prime

Sequence A001913 in Sloane's OEIS gives the sequence.

> Related question: what is rho(M,P) (Katz notation), the density of
> M _in_P_? I don't know the answer to this one.


The Wikipedia page gives that the conjectured value of rho(M,P) is
something called "Artin's constant," C_Artin = 0.3739558136...

http://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots



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