[seqfan] Re: A100083

[Rob Arthan]

The direct product of all the p-adic numbers contains all sorts of things David probably doesn't want, e.g., the elements e_p where e_p is 1 in the p-adic factor and 0 in all the others. Then e_2 and e_3 are non-zero but  e_2e_3 = 0 (zero divisors are inevitable in a direct product of non-trivial rings).

I think what David is interested in for given p, is the set, A_p say, of numbers that can be represented as limits of integer sequences that are convergent under the q-adic metric for all primes q. A_p is a subring, I think.  I imagine that A_p and A_q will not be isomorphic in general, as they are constructed by imposing quite different equivalence relations on the representing sequences. I don't know what is known about these rings.

Regards,

Rob. 

On 28 Jul 2013, at 02:54, franktaw at netscape.net wrote:

> 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.
>> 
>> 
>> _______________________________________________
>> 
>> Seqfan Mailing list - http://list.seqfan.eu/
>> 
>> 
>> 
>> _______________________________________________
>> 
>> Seqfan Mailing list - http://list.seqfan.eu/
> 
> 
> _______________________________________________
> 
> Seqfan Mailing list - http://list.seqfan.eu/
> 
> 
> _______________________________________________
> 
> Seqfan Mailing list - http://list.seqfan.eu/