 mersenneforum.org > Math A tentative definition
 Register FAQ Search Today's Posts Mark Forums Read  2018-10-23, 04:54   #12

May 2004

22·79 Posts Quote:
 Originally Posted by devarajkandadai Let N be a squarefree composite number with r factors, p_1,...p_r. Then we can define N as a tortionfree number if atleast two of its factors are inverses mod(P),where P is a prime number less than the largest prime factor of N.
We can now go to the next level. Note that 3 and 17 ( which are prime factors of the Carmichael number 561) are not only inverses (mod 5) but also inverses (mod 5^2).
(to be continued).   2018-10-24, 04:47   #13

May 2004

13C16 Posts A tentative definition

Quote:
 Originally Posted by devarajkandadai We can now go to the next level. Note that 3 and 17 ( which are prime factors of the Carmichael number 561) are not only inverses (mod 5) but also inverses (mod 5^2). (to be continued).
We can now define tortion free composite numbers as follows:
Let N be a squarefree composite number such that atleast two of its prime factors are inverses (mod P^k) where k is a natural number. Then N is a tortion free number of degree k. (P is a prime number less than the largest prime factor of N).

Last fiddled with by devarajkandadai on 2018-10-24 at 04:51 Reason: To make it clearer   2018-10-24, 05:51   #14

May 2004

22×79 Posts A tentative definition

Quote:
 Originally Posted by devarajkandadai We can now define tortion free composite numbers as follows: Let N be a squarefree composite number such that atleast two of its prime factors are inverses (mod P^k) where k is a natural number. Then N is a tortion free number of degree k. (P is a prime number less than the largest prime factor of N).
Example: 11305(ref a 104017) is tortion free of degree 2.   2018-10-24, 11:04   #15

May 2004

22×79 Posts Quote:
 Originally Posted by devarajkandadai We can now define tortion free composite numbers as follows: Let N be a squarefree composite number such that atleast two of its prime factors are inverses (mod P^k) where k is a natural number. Then N is a tortion free number of degree k. (P is a prime number less than the largest prime factor of N).
This class of composite numbers will be called "Madhavan numbers" .   2018-10-24, 12:26   #16
science_man_88

"Forget I exist"
Jul 2009
Dartmouth NS

841810 Posts Quote:
 Originally Posted by devarajkandadai This class of composite numbers will be called "Madhavan numbers" .
wouldn't powers of this form allow for Beal's conjecture solutions ?   2018-10-24, 13:45   #17
science_man_88

"Forget I exist"
Jul 2009
Dartmouth NS

2·3·23·61 Posts Quote:
 Originally Posted by science_man_88 wouldn't powers of this form allow for Beal's conjecture solutions ?
sorry still thinking in the wrong inverses   2018-10-24, 20:34   #18
CRGreathouse

Aug 2006

5,987 Posts Quote:
 Originally Posted by devarajkandadai We can now define tortion free composite numbers as follows: Let N be a squarefree composite number such that atleast two of its prime factors are inverses (mod P^k) where k is a natural number. Then N is a tortion free number of degree k. (P is a prime number less than the largest prime factor of N).
I have an example with degree k = 4031399, can anyone do better?   2018-10-28, 05:39   #19

May 2004

22·79 Posts A tentative definition

Quote:
 Originally Posted by devarajkandadai This class of composite numbers will be called "Madhavan numbers" .
Time to define inverses of higher order: let x and y be such that xy+1 = a*p^k+1 where a is a constant belonging to N, k is a natural number and p is a prime number. Then x and y are inverses of degree k.   2018-10-29, 12:33   #20

May 2004

13C16 Posts A tentative definition

Quote:
 Originally Posted by CRGreathouse I have an example with degree k = 4031399, can anyone do better?
Charles, pl give details of above number.   2018-10-29, 14:41 #21 Batalov   "Serge" Mar 2008 Phi(4,2^7658614+1)/2 89×113 Posts Hint: P=2, and P^k+1 is a semiprime.   2018-10-29, 15:13   #22
Dr Sardonicus

Feb 2017
Nowhere

2·11·283 Posts Quote:
 Originally Posted by CRGreathouse I have an example with degree k = 4031399, can anyone do better?
While rummaging around on line, I blundered into the candidates k = 13347311 and k = 13372531 due to R. Propper (Sep. 2013).   Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post devarajkandadai Number Theory Discussion Group 10 2018-07-22 05:38 lfm PrimeNet 4 2009-11-15 00:43 R.D. Silverman Math 47 2009-09-24 05:23 Greenbank Octoproth Search 4 2007-12-07 18:41 Damian Lounge 1 2007-05-27 13:30

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