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

[jean-paul allouche]

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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