A little number theory....
Expressed in base 10 (or anything) notation, rational numbers have a
repeating
pattern of _fixed_ length. (Sometimes the repeating digit pattern is '0'
as in .50 for 1/2
in base 10 -- but this gets very close to a quibble).
If n is relatively prime to 10 (not divisible by 2 or 5), and if phi(n) is
the 'totient' function
or the number of integers between 1 and n that are relatively prime to n,
then
10^phi(n) has residue (reste) 1 on division by n or
10^phi(n) = 1 (modulo n).
This is a variant on the Little Fermat Theorem and the Chinese Remainder
Theorem
or the group of units in a finite ring (or several other ways of
constructing the same thing).
For example, 10^12 = 13k + 1.
Actually, 10^6 = 1 + 13*76923. The period of 1/13 is 6.
We are guaranteed to get a repeating pattern for 1/n of length dividing
phi(n).
Now, phi(9) = 4 and the period of 1/9 is just 1 so phi(n) is a maximum
period length.
Back to pi: pi is not just irrational, it is transcendental. The square
root of 2 is irrational
but is a solution to X^2 - 2 = 0. Numbers which solve some polynomial (with
integer coefficients) are called
algebraic. Numbers not algebraic are called transcendental.
JT
----- Original Message -----
From: Joth Tupper <[EMAIL PROTECTED]>
To: GIMPS <[EMAIL PROTECTED]>
Sent: Thursday, October 21, 1999 7:33 AM
Subject: Re: Mersenne: RE: PI and other periods
> Anyone remember Louisville numbers?
>
> The simplest I recall is .110001000000000000000001000000... = sum of
> 0.1^(n!) for n=1,2,3,...
>
> Many such numbers and constructions exist which share strange properties:
>
> 1) the digit patterns exist and are well-defined
> 2) the numbers, like pi, are transcendental [i.e., cannot be roots of any
> polynomial in one variable with integer coefficients].
>
> [Recall that the rational number p/q is a root of the "polynomial" qX -
p
> = 0.]
>
> Showing pi transcendental takes a lot of effort. Showing the Louiville
> numbers transcendental
> takes a lot less effort (but maybe a lot more memory than I seem to
have!).
>
> Joth
>
>
>
> ----- Original Message -----
> From: Philippe Trottier <[EMAIL PROTECTED]>
> To: <[EMAIL PROTECTED]>
> Sent: Wednesday, October 20, 1999 10:39 PM
> Subject: Mersenne: RE: PI and other periods
>
>
> > HI,
> >
> > Again if you look from a human eye, we can see (imagine) a nearly
possible
> > period in that number ..., again that's human brain doing overtime...
But
> > again MAYBE, there is a real period to that number... and this number
also
> > start to have a considerable amount of known digits (We would have to
> share
> > multiple generation just to say it)
> >
> > Philippe
> >
> > At 12:09 20.10.1999 +0100, you wrote:
> > >Value of pi is the product of the infinite series ...
> > >
> > >pi = 4 (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 -1/15 + 1/17.........)
> > >
> > >Hope this helps..
> > >
> > >Regards,
> > >Ian McLoughlin, Chematek U.K.
> > >
> > >Tel/Fax : +44(0)1904 679906
> > >Mobile : +44(0)7801 823421
> > >Website: www.chematekuk.co.uk
> >
> > _________________________________________________________________
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> > Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
> >
>
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