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

[Max Alekseyev]

Catalan's conjecture (and its generalizations) is an overkill here. When
power bases are fixed, the problem becomes much easier. E.g., one does not
need Mihailesu's proof (which is rather complicated) to find all solutions
to 2^x - 3^y = 1 or -1 as this equation can be solved directly by
elementary means.

Regards,
Max
On May 26, 2014 10:36 AM, "jean-paul allouche" <
jean-paul.allouche at imj-prg.fr> 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
>>>
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