- 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
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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