A tentative definition

[devarajkandadai]

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).

[devarajkandadai]

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).

[devarajkandadai]

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.

[devarajkandadai]

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" .

[science_man_88]

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 ?

[science_man_88]

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

[CRGreathouse]

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?

[devarajkandadai]

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.

[devarajkandadai]

Originally Posted by CRGreathouse

I have an example with degree k = 4031399, can anyone do better?

Charles, pl give details of above number.

[Batalov]

Hint: P=2, and P^k+1 is a semiprime.

[Dr Sardonicus]

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).