> ----- Original Message -----
> From: Bob Margulies <[EMAIL PROTECTED]>
> To: <[EMAIL PROTECTED]>
> Sent: Sunday, October 24, 1999 9:15 AM
> Subject: Mersenne: More Schlag
>
>
> In reading the numerous postings about Liouville numbers and patterned
> transcendentals, I notice that there has been a careful avoidance of the
> definition of the term 'pattern.' If the Liouville Transcendental Number
> is expressed in a base other than 10, I would guess that there's still a
> pattern, but I don't know how to see it. Perhaps a pattern is something
> that sets the 'I see a pattern' bit in my head.
Sounds ok by me.
I would accept as a 'pattern' for a (possibly transcendental) number ANY
closed
form expression giving a general term. That is, there is some function
f(n)
for (positive)
integers n such that a number with 'pattern' is given by the infinite sum
f(1)+f(2)+f(3)+...+f(n)+...
The Liouville number I remembered was
f(n) = (0.1)^(n!) for n=1,2,3,...
A simpler (and larger) number used
f(n) = (0.1) ^ (n^2)
As I recall, either of these definitions can use any rational number
(between 0 and 1) in place
of 0.1 and we get a transcendental number. Using something convenient in
base 10 is not the critical
point. The critical question is the form of f(n) -- in this case the kind
of exponent grows faster than a
linear function.
I seem to recall a non-intuititive theorem about rational approximations to
numbers (this is from c. 1968).
If you can approximate a number too closely, then it is transcendental.
S.Lang wrote a book on
trancendental numbers and degrees around 1973 and a precise statement might
be there.
Does anyone recall this?
JT
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