[seqfan] Re: A100083

[franktaw at netscape.net]

You're thinking too narrowly. It's the direct product: you can take 
whatever number you want in the p-adics for each p. So there's a value 
that is sqrt(17) in the 2-adics, -1/2 in the 3-adics, some uncomputable 
value in the 5-adics, etc.

If you look at more closely, I think you'll see that your idea of a 
"number" in this context is incoherent.

Franklin T. Adams-Watters

-----Original Message-----
From: David Wilson <davidwwilson at comcast.net>

There are some numbers that exist in all the p-adics, e.g. the 
integers, and
the F I described.

However, there are some numbers that exist in some but not all p-adics.
An example would be x = ...11112 in the 3-adics, because it solves 2x = 
1,
but this latter equation has no solution in the 2-adics, so x has no 
2-adic
equivalent.

So the set of numbers I describe seems to be a subset of the 2-adics 
(or the
p-adics for any p) which is homomorphic to a subset of each of the other
p-adics.
The homomorphism in question is an identity on the integer, and on F.

Specifically, the set of numbers I subset of the 2-adics, which is 
nowhere
near as large as the direct product of all p-adics.

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of
> franktaw at netscape.net
> Sent: Saturday, July 20, 2013 3:26 PM
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Re: A100083
>
> Sorry, that should be direct product, not direct sum.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: franktaw <franktaw at netscape.net>
>
> This class can characterized as the direct sum of the p-adic integers 
for
all
> primes p. It is rather interesting; for one thing, it is isomorphic 
to the
> endomorphisms of the torsion group Q/Z (where this is understood as
> referring to the additive groups of these rings).
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: David Wilson <davidwwilson at comcast.net>
>
> To me, it looks as if there is a generalization of the integers to a
broader class
> of numbers (like F) that have p-adic representations for all primes p.
> This latter class of numbers seems to have some interesting 
divisibility
> properties.
>
> Since I have only a very tenuous grasp of p-adic theory, I have no 
idea
how to
> develop this idea.
>
>
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