[seqfan] Re: A100083

[franktaw at netscape.net]

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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