[seqfan] Re: A100083

David Wilson davidwwilson at comcast.net
Thu Aug 15 04:57:20 CEST 2013 

I just discovered that my Outlook mail reader has been selectively
spaminating seqfan emails.
No wonder the threads have seemed disjointed.
Among the victims were several mails on this subject, so I didn't see the
attached message until I perused my spam folder.

At any rate, good proof Franklin.

I had realized that [1] was wrong, but no so entirely wrong that you have
shown it to be.
My proof was as follows:

There is clearly an integer sequence that converges to p-adic 0 for every p,
namely the constant 0 sequence.

Now consider the sum of sequence S = A006939, the running product of the
primorials (indexed starting at 0):

1 + 2 + 12 + 360 + 75600 + ...

S converges in every p-adic so S is panadic.

Note also that 2^n is the largest power of 2 dividing S(n).

Now define a new integer sequence T as follows:

	T(0) = 1.
	T(n+1) = T(n) + S(k) where 2^k is the largest power of 2 dividing n.

So T gives the partial sums of a subsequence of S

	1 + S(0) + S(1) + S(2) + S(4) + ... = 1 + 1 + 2 + 12 + 75600 + ...

where each term is chosen to increase the power of 2 dividing the partial
sum.
Since S is panadic, every subsequence is panadic, and T converges in every
p-adic.
Since the powers of 2 in the partial sums of T increase without bound, T
converges on 2-adic 0.
But starting at the 12 term, every subsequent term is divisible by 3, so T
converges to a 3-adic number ending in 1, that is, not 0.

We have two sequences (the constant zero sequence and T) which converge to
2-adic 0, but to different 3-adic numbers.

That was enough to thwart [1], but you completely blew it to shreds.


> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of
> franktaw at netscape.net
> Sent: Sunday, July 28, 2013 5:03 PM
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Re: A100083
> 
> But your [1] is not true. It is about as far from being true as one can
imagine.
> You can choose an arbitrary p-adic integer c_p for each p, and construct a
> sequence S which converges to c_p in the p-adic integers for every prime
p.
> 
> Let me give an example: take c_2 = 1, and c_p = 0 for every other p.
> Now start looking at factorials:
> 
> 2! = 2, so we need 1 (mod 2); a(2) = 1
> 3! = 2 * 3, we need 1 (mod 2) and 0(mod 3) => 3 (mod 6); a(3) = 3 4! = 2^3
* 3,
> we need 1 (mod 8) and 0 (mod 3), giving 9 (mod 24); a(4) = 9 5! = 2^3 * 3
* 5;
> so we need 1 (mod 8), 0 (mod 3) and 0 (mod 5) => 105 (mod 120); a(5) = 105
> 6! = 2^4 * 3^2 * 5; 225 == 1 (mod 16), 0 (mod 9) and 0 (mod 5); so a(6) =
225 ...
> 
> So the desired sequence starts 1, 3, 9, 105, 225, ...
> (Arguably there should be an initial 0 there, but that is not relevant to
the
> current discussion.)
> 
> Franklin T. Adams-Watters
> 
> -----Original Message-----
> From: David Wilson <davidwwilson at comcast.net>
> 
> 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.
> > >>
> > >>
> > >> _______________________________________________
> > >>
> > >> Seqfan Mailing list - http://list.seqfan.eu/
> > >>
> > >>
> > >>
> > >> _______________________________________________
> > >>
> > >> Seqfan Mailing list - http://list.seqfan.eu/
> > >
> > >
> > > _______________________________________________
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> >
> >
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> >
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> 
> 
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