[seqfan] Re: A100083

[Rob Arthan]

I now agree. Every p-adic integer can indeed be represented by a sequence that converges in the q-adic metric for every prime q.

It would be nice to know if there is a name for the ring you have been discussing (the completion of the integers with respect to a metric in which the subgroups of Z comprising multiples of k! for large k have small radii for large k). Thank you for drawing this interesting ring to my attention.

On 28 Jul 2013, at 18:57, franktaw at netscape.net wrote:

> As you described it, it is. I showed you that it is. If you think I made a mistake, please tell us where,
> 
> Franklin T. Adams-Watters
> 
> -----Original Message-----
> From: Rob Arthan <rda at lemma-one.com>
> 
> 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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