[seqfan] Re: A100083

[David Wilson]

Yes, I think Rob is getting closer to what I'm thinking.

Suppose a sequence of integers S has a p-adic limit for each prime p. Call S
a "panadic" sequence.

One would hope that

[1] If two panadic sequences S and T have the same p-adic limit for any
prime p, then they have the same p-adic limit for each prime p.

If [1] is true, one can then define a "panadic number" P as the set of all
panadic sequences (of integers) having the same p-adic limit for some
specific prime p (say, 2-adic limit). [1] then gives that each element of P
has the same p-adic limit for any p, meaning that the panadic numbers are
well defined.

I believe [1] would also imply that if two panadic sequences S and T have
distinct p-adic limits for any prime p, they have distinct limits for each
prime p. This would mean that each panadic number has a unique
representation in each of the p-adics, and would induce an equivalence
relation between these representations.

For example, let's return to F = 0! + 1! + 2! + ....  The sequence S of
partial sums converges to a unique F_p in each of the p-adics. Given [1],
this would make F a panadic number. Presumably, any sequence of integers
converging to F_p in the p-adics for any specific p would converge to F_p in
the p-adics for all p.  For any p and q, this would induce a bijection
between the p-adic and q-adic representations of the panadic numbers.
Hopefully, this bijection would be a homomorphism, preserving addition,
multiplication, &c.

Some integer sequences converge in the p-adics for some p, but not others.
For example, take the sequence S = {(3^n+1)/2}, which converges to ...11112
in the 3-adics but does not converge in the 2-adics.  S therefore does not
define a panadic number. If [1] is true, then any sequence converging to
...11112 in the 3-adics would likewise diverge in the 2-adics, so that
...11112 in the 3-adics does not represent a panadic number. This means the
panadic limits in the 3-adics are a strict subset of the 3-adics as a whole,
an observation easily generalized to each of the p-adics.

Each integer n is panadic, since the constant sequence {n} converges in the
p-adics for all p. Thus the integers are all panadic numbers.

The question now becomes, is [1] in fact true, and if so, what is the
structure of the panadic numbers?


> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Rob
> Arthan
> Sent: Sunday, July 28, 2013 10:13 AM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: A100083
> 
> 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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