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

[Benoît Jubin]

Take A = p^a and B = q^a in the identity
A^b - B^b = (A-B) \sum_{i=1}^{b-1} A^i 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/
>> >
>>
>> _______________________________________________
>>
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>>
>
>
>
> --
> 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/