----- Original Message -----
>From: Jud McCranie
>To: Chip Lynch
>Cc: GIMPS Project Mailing List
>Sent: Friday, October 22, 1999 6:59 AM
>Subject: Re: Mersenne: Re: Schlagobers, Louisville style
>At 11:47 PM 10/21/99 -0400, Chip Lynch wrote:
>> There's no reason for it to be COMPLETELY random. Plenty of
trancendental
>>numbers have easy to view patterns
>Only an infinitesimally small % of them have patterns. For each one that
has a pattern there are an infinite number that don't.
Gotta watch those infinities. For each (positive) even integer we can
construct an infinite set of odd numbers to go with it. Here is one way:
Let 2k > 0 be given. let P(k) = k'th odd prime (so P1 = 3, P2 = 5, ...).
Consider the set of integers S(k) = {P(k), P(k)^2, P(k)^3,...}.
Clearly S(k) is infinite, and if j and k are different, then S(k) and S(j)
are disjoint.
So what?
Now, back to Jud's comment.
Well, of course! There are uncountably many transcendentals because there
are uncountably many real numbers but only countably many algebraic numbers.
Unless someone can suggest an infinite (I think we need uncountable)
collection of patterns for transcendental numbers,
the collection of transcendental numbers with patterns will be countable.
With infinitely many (but countable) patterns,
it is possible to construct 2^(aleph-null) transcendentals with patterns
(of some sort) but only using infinite series -- which may
turn up algebraic numbers in the limits. Finite linear (over rationals)
combinations of transcendentals will be transcendental but
this will keep the set with patterns countable.
A countable subset of real or complex numbers has measure 0 in the usual
measure
(Lebesgue or Haar or any other derivation where the length of an interval on
the real line is its measure
or the derived measure in the plane based on the product topology).
Sets of measure 0 represent 0% probabilities in any Borel probability
measure based on intervals.
On the other hand, remembering Littlewood, I think, finitely many instances
of transcendental
numbers with patterns will give us what we need for all practical purposes.
Having countably many available sounds like 'plenty' to me.
JT
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