Let $\pi(n)$ be the prime counting function. And let $\varphi(n)$ be the count of Carmichael pseudo-primes for $1$ to $n$. Is the ratio,
is known, as $n \to \infty$? I am asking this because of i want to know how much safe is the Fermat Primality Test (the one which only uses Fermat's Little Theorem).
I recently wrote a Python script (initially for RSA) and produced more than 10, random 1024 bit primes with the help of this test. After i used the powerful mathematical software PARI/GP to check if they are really primes or pseudo-primes. In conclusion none of them were pseudo-primes. So, if know this rate, i will know the probability that this test will encounter a pseudo-prime and fail.