If the new primes appeared one at a time, it would make the task of Factoring Mersenne numbers much easier: To factor Mn, just test for & Remove all the earlier divisors of Msmaller; any leftover piece would be Prime. No such luck. To build on the 2^11-1 = 23.89 example ... 4^11-1 = 2049.2047 = 3.683.23.89, so the sequence (4^N-1)/3 will have First appearances of 23, 89, and 683 at term N=11. A similar pattern arises for 4^Prime-1, where all the new prime divisors Are of the shape 2 K Prime + 1. (23 & 89 are examples with Prime=11.) Rich -----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Allan Wechsler Sent: Thursday, July 23, 2015 12:38 PM To: math-fun Subject: [EXTERNAL] Re: [math-fun] Number theory pattern? s_3 = 21 introduces 3 and 7 -- isn't that an example of Dan's (2)? s_5 = 341 = 11*31; isn't that another example of two primes making a simultaneous debut? On Thu, Jul 23, 2015 at 2:10 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
1) Yes, that list is by induction the list of primes that divide any s_n, and I've shown that every (odd) integer divides infinitely many s_n. Hence every odd prime occurs as a primitive prime divisor.
2) I see no reason why that shouldn't happen. (Indeed, if that weren't the case, then by (1) we would have some bizarre enumeration of the primes.)
Sent: Thursday, July 23, 2015 at 6:29 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Number theory pattern?
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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