(1) Yes. Suppose you want to know the first element A002450 that is divisible by 59. Calculate the powers of 4 mod 59. They go: 1, 4, 16, 5, 10, ... and eventually they must cycle back to 1 because of general Fermat stuff. Certainly 4^58 = 1 mod 59. So 4^58-1 is a multiple of 59, and so is (4^58-1)/3. (2) My intuition is yes. Certainly in other similar sequences (the Mersenne numbers, for example) you can get multiple novel primes. 2^11-1, for instance, is 23*89. I don't know how to construct an example for A002450, though. On Thu, Jul 23, 2015 at 1:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks for all the fascinating stuff I knew nothing about.
Incidentally, the list of primitive prime divisors (primes that don't divide any previous term of s_n = (4^n-1)/3) in A129735 raises a question or two whose answers I may have missed:
1) Does every prime appear in that list? (A129735 seems to include every prime through at least 53.)
2) Can a new term ever include more than one primitive prime factor?
—Dan
On Jul 23, 2015, at 8:01 AM, Neil Sloane <njasloane@gmail.com> wrote:
The primitive divisors of A002450 are listed in A129735 (along with a reference to Zsigi)
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Jul 23, 2015 at 8:41 AM, Victor S. Miller < victorsmiller@gmail.com> wrote:
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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