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Old 2018-10-23, 04:54   #12
devarajkandadai
 
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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).
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Old 2018-10-24, 04:47   #13
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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
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Old 2018-10-24, 05:51   #14
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Default A tentative definition

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Originally Posted by devarajkandadai View Post
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.
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Old 2018-10-24, 11:04   #15
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Originally Posted by devarajkandadai View Post
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" .
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Old 2018-10-24, 12:26   #16
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This class of composite numbers will be called "Madhavan numbers" .
wouldn't powers of this form allow for Beal's conjecture solutions ?
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Old 2018-10-24, 13:45   #17
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wouldn't powers of this form allow for Beal's conjecture solutions ?
sorry still thinking in the wrong inverses
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Old 2018-10-24, 20:34   #18
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Quote:
Originally Posted by devarajkandadai View Post
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?
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Old 2018-10-28, 05:39   #19
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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.
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Old 2018-10-29, 12:33   #20
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I have an example with degree k = 4031399, can anyone do better?
Charles, pl give details of above number.
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Old 2018-10-29, 14:41   #21
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Hint: P=2, and P^k+1 is a semiprime.
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Old 2018-10-29, 15:13   #22
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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).
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