Conjecture (contd).

This pertains to my conjecture that a ll the factors of a Carmichael number

cannot be Mersenne.

What folows is a heuristic.. Let us first consider 3-factor Cns. The

implication of Pomerance's

proof: (attachment to my previous post): If P_1,P_2 & P_3 are the three

factors of a CN ( say N) then

(P_1-1)*(N-1/(P_2-1)*(P_3-1) = k is an integer. Now let, IF POSSIBLE, (

2^p_1), (2^p_2-1) & (2^p_3-1) be the three Mersenne prime factors of N, a

Carmichael number.

Then it follows from Fermat's theorem that 2^p_1-2 is a multiple of p_1.

Similar remarks apply to

p_2 and p_3.

The implication is that k above must have p_2 and p_3 as factors of the

denominator of k. Considering that (P_2-1)*(N-1)/P_1-1)*(P_3-1) should also

be an integer in the case of Carmichael

numbers the implication is that p_1 must also be a factor of the denominator

of k above.

There are also other unspecified prime factors in the denominator. If should

be an integer (N-1) in the numerator of k must also have all these prime

factors which is highly improbable. This improbabality grows exponentially

as the number of factors of a CN grow. Hence my conjecture,

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