These are called primitive divisors. There's a theorem of Zsigismondy which covers this . See this paper for details: http://www.uea.ac.uk/~h008/research/primes.pdf Victor Sent from my iPhone
On Jul 22, 2015, at 18:22, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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