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6:33 PM
Hey @SEJPM
I am reading about Carmichael numbers. And I have a question... Could I maybe ask you?
 
@Evinda sure
 
I haven't understood why the last proposition holds. Why isn't it possible to make $\Omega(p_0)$ repetitions, given that the size of $p_0$ is smaller than this of n ? @SEJPM
 
@Evinda I think it is because if a carmichael number has only few and prime factors, randomly finding them or multiples of them is hard
and nobody will try to factor a number for which they want to prove primality
and the defining property of a carmichael number is to pass the Fermat test for all bases that are co-prime with the number
so showing non-primality of a carmichael number (using the Fermat test) is exactly as hard recovering a non-trivial factor
 
And we say that it is hard when $p_0$ has more than 20 decimal digits, because it is difficult to find a non-trivial factor of n, when its smallest divisor has more than 20 digits? @SEJPM
 
6:49 PM
@Evinda yes, the largest factor found using "small-factor factorization" had like 80- bits digits
also common primality testing algorithm will try divisions by small primes and 20-digits are certainly out of range for that
 
 
2 hours later…
8:21 PM
Ah I see... Thank you very much :) @SEJPM
 
:)
 

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