[seqfan] Re: A100083

[franktaw at netscape.net]

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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