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

[eric41293 at comcast.net]

If n = ab then k^n - (k-1)^n = (k^a)^b - ((k-1)^a)^b is
divisible by k^a - (k-1)^a by the usual "difference of powers" formula.

----- Original Message -----
From: "Alonso Del Arte" <alonso.delarte at gmail.com>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Sent: Wednesday, September 11, 2013 9:22:53 PM
Subject: [seqfan] Prove that these exponents are primes (e.g., A062608)

Lately, I've been working on a little simplification of several sequences
of 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/