[seqfan] Re: A100083

[Rob Arthan]

The ring I have called A_p is not the same as the ring of p-adic integers. 

On 28 Jul 2013, at 15:41, franktaw at netscape.net wrote:

> What you describe is the same thing.
> 
> The ring I'm talking about can also be described as the generalization of the base factorial representation of the integers, in the same way that the p-adic and g-adic numbers are the generalization of the base p or base g  numbers. I.e., each can be represented (uniquely)  as
> 
>   sum(k=1..infinity, c(k)*k!)
> 
> where 0 <= c(k) <= k. David's number F was the case where c(k) = 1 for every k.
> 
> Now the numbers
> 
>   a(n) = sum(k=1..n, c(k)*k!)
> 
> converge to the specified p-adic integer for each p.
> 
> If you focus only on a particular p, as in your A_p, you are pulling out the p-adic component from the direct product, and will get simply the p-adic integers.
> 
> Franklin T. Adams-Watters
> 
> -----Original Message-----
> From: Rob Arthan <rda at lemma-one.com>
> 
> 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.
>>> 
>>> 
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