@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

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

@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

Ah I see... Thank you very much :) @SEJPM

:)