[seqfan] Re: A006899, A108906 and similar sequences : divergence proof?

[Bernard Vatant]

Hi all

Thanks all for all those helpful answers and related papers. Jean-Paul, I
now have reading for quite a couple of nights :)
"a^k - b^n = c has only finitely many solutions" (a and b being
multiplicatively
independent integers) seems to settle the case for integers.

For any integer N (arbitrarily great)
|a^k - b^n| = c has a finite number of solutions for any c < N, and the set
of all those k,n solutions for all values of c < N is finite, hence has an
upper bound M. If k,n > M then |a^k - b^n| > N.
Hence A108906 is divergent, as well as all similar difference sequences.

But not sure we can extend this result to multiplicatively independent
*rationals*

Remember my initial question was about powers of 2 and 3/2 ...

Cheers

Bernard

2014-05-26 16:51 GMT+02:00 <israel at math.ubc.ca>:

> Pillai's conjecture is a much stronger statement. But your reference
> mentions a theorem of Pillai that gives asymptotics on the number of
> integer solutions (x,y) of 0 <= a^x - b^y <= c for any fixed a, b >= 2. A
> consequence of this is that a^x - b^y = c has only finitely many solutions.
>
> Cheers,
> Robert
>
>
> On May 26 2014, jean-paul allouche wrote:
>
>  Hi The statement for n = 1 is a particular case of Catalan's conjecture,
>> [proven by Preda Miha ̆ilescu, see, e.g., the expository paper
>> http://archive.numdam.org/ARCHIVE/SB/SB_2002-2003__45_/
>> SB_2002-2003__45__1_0/SB_2002-2003__45__1_0.pdf ] The geenral case is
>> known as Pillai's conjecture, see, e.g., http://hal.upmc.fr/docs/00/40/
>> 51/19/PDF/PerfectPowers.pdf
>>
>> best
>> jpa
>>
>>
>> Le 26/05/14 16:05, Max Alekseyev a écrit :
>>
>>> The divergence of A108906 is equivalent to the statement that for each
>>> integer n, the equation 2^x - 3^y = n has only a finite number of
>>> integer solutions (x,y).
>>> It is easy to prove this statement for each particular (given) value
>>> of n and even compute all the solutions, but I do not know how to
>>> prove it for general n.
>>>
>>> Regards,
>>> Max
>>>
>>>
>>>
>>> On Fri, May 23, 2014 at 7:26 PM, Bernard Vatant <bvatant at gmail.com>
>>> wrote:
>>>
>>>> Hello sequence fans
>>>>
>>>> I'm new to this list and the archives are huge, so please point me to
>>>> any previous message if this has been answered. I've been struggling with
>>>> sequences formed by merging two geometric progressions, the simpler being
>>>> A006899, merging powers of 2 and 3, and particularly in the sequence of
>>>> differences between its successive terms, such as A108906. Although it
>>>> seems highly "obvious" by looking at the 1000 first terms of A108906 that
>>>> it should be divergent, I could not find any proof of that divergence, nor
>>>> even a statement that it is indeed divergent. I have a strong conjecture
>>>> for any real numbers p and q such as 1 < p < q and p^k != q^n for all
>>>> integers k,n Define the increasing sequence u(n) containing both powers of
>>>> p or q on the model of A006899, and its differences sequence
>>>> d(n)=u(n+1)-u(n) on the model of A108906. My conjecture is that d(n) is
>>>> divergent ... I would be happy to have a proof for p and q integers, and
>>>> even for p=2 and q=3.
>>>>
>>>> Thanks for any clue, related works etc.
>>>>
>>>> Bernard
>>>>
>>>