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

jean-paul allouche jean-paul.allouche at imj-prg.fr
Mon May 26 16:51:55 CEST 2014


Sure. But Bernard Vatant mentioned: "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..."

best
jp


Le 26/05/14 16:48, Max Alekseyev a écrit :
> 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 general case is known as Pillai's conjecture, see, e.g.,
>> http://hal.upmc.fr/docs/00/40/51/19/PDF/PerfectPowers.pdf
>>
>> best
>> jpa
>>
>>
>>




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