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

[jean-paul allouche]

right! :-)
best
jp


Le 26/05/14 16:51, israel at math.ubc.ca a écrit :
> 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
>>>>
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