[seqfan] Re: Prove that these exponents are primes (e.g., A062608)

[Benoît Jubin]

> A^b - B^b = (A-B) \sum_{i=1}^{b-1} A^i B^{b-i}

This should of course read A^b - B^b = (A-B) \sum_{i=1}^{b} A^{i-1} B^{b-i}.


>
> (but you don't need the RHS for the proof: just say that the
> polynomial (in A) A^b - B^b vanishes when A=B, therefore is a multiple
> of A-B).
>
>
> On Mon, Sep 16, 2013 at 3:37 PM, Alonso Del Arte
> <alonso.delarte at gmail.com> wrote:
>> Dear Susanne,
>>
>> It is wrong. I need to correct it to something like k^(n - a) - 2k^n/(k -
>> 1)^a + (k - 1)^(n - a). I believe that's correct but it can be simplified
>> further.
>>
>> Al
>>
>>
>> On Mon, Sep 16, 2013 at 1:33 AM, Susanne Wienand
>> <susanne.wienand at gmail.com>wrote:
>>
>>> Dear Alonso,
>>>
>>> The equation in the proof on
>>> https://oeis.org/wiki/Primes_as_differences_of_powers seems to be wrong.
>>>
>>> If I plug in n = 12, a = 3, b = 4 and k = 5, I get for the left side:
>>>
>>> (5^12 - 4^12) / (5^3 - 4^3) = 3727269
>>>
>>> and for the rigth side:
>>>
>>> 5^9 - 4^9 = 1690981
>>>
>>> Regards
>>> Susanne
>>>
>>>
>>> 2013/9/13 Alonso Del Arte <alonso.delarte at gmail.com>
>>>
>>> > Thank you very much, Robert, Eric, Vladimir.
>>> >
>>> > I don't intend to add a comment to that effect to almost a hundred
>>> entries
>>> > in the OEIS, but I have added it to this OEIS Wiki page:
>>> > https://oeis.org/wiki/Primes_as_differences_of_powers
>>> >
>>> >
>>> > On Thu, Sep 12, 2013 at 5:05 AM, Vladimir Shevelev <shevelev at bgu.ac.il
>>> > >wrote:
>>> >
>>> > > If n is not prime, e.g., n=p*q , p>1 is prime, then 42^(p*q) - 41^(p*q)
>>> > >  is multiple of 42^p - 41^p.
>>> > >
>>> > > Regards,
>>> > > Vladimir
>>> > >
>>> > > ----- Original Message -----
>>> > > From: Alonso Del Arte <alonso.delarte at gmail.com>
>>> > > Date: Wednesday, September 11, 2013 15:23
>>> > > Subject: [seqfan] Prove that these exponents are primes (e.g., A062608)
>>> > > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>>> > >
>>> > > > Lately, I've been working on a little simplification of several
>>> > > > sequencesof numbers such that k^n - (k - 1)^n is prime. A few of
>>> > > > these entries
>>> > > > contain a remark to the effect that "all terms are prime," but
>>> > > > this is
>>> > > > stated without proof. The most famous case is of course that of the
>>> > > > exponents for the Mersenne primes, k = 2. The proof that the
>>> > > > primality of n
>>> > > > is a necessary but not sufficient condition is well-known and
>>> > > > simple enough.
>>> > > >
>>> > > > It seems simple to extend this to all k, but the proof has
>>> > > > eluded me. First
>>> > > > I thought it would be a simple application of Fermat's little
>>> > > > theorem. Then
>>> > > > I thought it was just a matter of generalizing the proof for k =
>>> > > > 2. Any
>>> > > > thoughts?
>>> > > >
>>> > > > Al
>>> > > >
>>> > > > --
>>> > > > Alonso del Arte
>>> > > > Author at
>>> > > > SmashWords.com<https://www.smashwords.com/profile/view/AlonsoDelarte
>>> > >Musician
>>> > > at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>>> > > >
>>> > > > _______________________________________________
>>> > > >
>>> > > > Seqfan Mailing list - http://list.seqfan.eu/
>>> > > >
>>> > >
>>> > >  Shevelev Vladimir
>>> > >
>>> > > _______________________________________________
>>> > >
>>> > > Seqfan Mailing list - http://list.seqfan.eu/
>>> > >
>>> >
>>> >
>>> >
>>> > --
>>> > Alonso del Arte
>>> > Author at SmashWords.com<
>>> > https://www.smashwords.com/profile/view/AlonsoDelarte>
>>> > Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>>> >
>>> > _______________________________________________
>>> >
>>> > Seqfan Mailing list - http://list.seqfan.eu/
>>> >
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>
>>
>>
>> --
>> Alonso del Arte
>> Author at SmashWords.com<https://www.smashwords.com/profile/view/AlonsoDelarte>
>> Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/