A tentative definition

[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.

[science_man_88]

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.

6p with p prime and 1 mod 3 are tortion free
10p with p>3; p 3 mod 5
at least using additive inverses. multiplicayive inverses are different.

[BudgieJane]

What is tortion? Did you mean torsion?

[Batalov]

No, he meant torture. Seriously.

[Dr Sardonicus]

If "inverse" means "multiplicative inverse" we have the following:

If N is odd and the product of at least two prime factors, any two of its factors are multiplicative inverses (mod 2), and P = 2 is less than the largest prime factor of N.

If N has at least three prime factors, then at least two of them are multiplicative inverses (mod 2), and P = 2 is less than the largest prime factor of N.

(In the above two cases, of course, any odd factor is self-inverse (mod 2).)

This leaves N = 2*p, where p is prime. If N - 1 = 2*p - 1 is prime, it satisfies the definition. Otherwise, it does not.

In short, the numbers fitting the definition are of the form N = 2*p, where p > 2 is prime, and 2*p - 1 is also prime.

If you don't like the choice P = 2, you should have said so. However, even if you exclude P = 2 by fiat, your options are still limited:

If N has at least four prime factors if 3 divides N, or at least three prime factors if 3 does not divide N, then N has at least one pair of factors which are multiplicative inverses (mod 3), and P = 3 is less than the largest prime factor of N.

[devarajkandadai]

Originally Posted by science_man_88

6p with p prime and 1 mod 3 are tortion free
10p with p>3; p 3 mod 5
at least using additive inverses. multiplicayive inverses are different.

Rightm now refering only to multplicative inverses

[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.

Inspiration comes from group theory where normal sub-groups are tortion free.
Am refering only to multiplicative inverses.

[CRGreathouse]

Up to 100, I find: 10, 15, 21, 22, 26, 30, 33, 34, 35, 39, 42, 46, 51, 55, 57, 58, 65, 66, 69, 70, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95.

Up to a million there are 607926 squarefree numbers, of which 525128 meet your definition.

Asymptotically the fraction is 1.

Edit: I forgot to post, so Sardonicus beat me to it.

[science_man_88]

Originally Posted by Dr Sardonicus

In short, the numbers fitting the definition are of the form N = 2*p, where p > 2 is prime, and 2*p - 1 is also prime.

AKA prime p such that p-1 is not of form 2ij+i+j via the sieve of sundaram.

[devarajkandadai]

Originally Posted by CRGreathouse

Up to 100, I find: 10, 15, 21, 22, 26, 30, 33, 34, 35, 39, 42, 46, 51, 55, 57, 58, 65, 66, 69, 70, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95.

Up to a million there are 607926 squarefree numbers, of which 525128 meet your definition.

Asymptotically the fraction is 1.

Edit: I forgot to post, so Sardonicus beat me to it.

Asymtotically the fraction may be 1. Now I am able to conjecture
that a necessary condition for a squarefee composite ( with minimum 3 prime factors) to be a Devaraj number (which include Carmichael numbers) is that it should be tortion free.

[CRGreathouse]

Originally Posted by devarajkandadai

Asymtotically the fraction may be 1. Now I am able to conjecture
that a necessary condition for a squarefee composite ( with minimum 3 prime factors) to be a Devaraj number (which include Carmichael numbers) is that it should be tortion free.

A squarefree number with at least three prime factors has at least two distinct prime factors, which are multiplicative inverses mod 2.