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

israel at math.ubc.ca israel at math.ubc.ca
Mon May 26 16:51:30 CEST 2014


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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>>>
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