RSA primes vs. largest known primes In the context of a new largest (mersenne) prime number being found this week -

The largest known prime number is now 2^57,885,161 − 1, and it took 5 years to find it since the last largest prime was found.

But we know that various asymmetric encryption algorithms require ridiculously large primes which are used as the infamous p and q factors. For example, 1024-bit RSA would require two 512-bit primes.

But this article, if correct, claims we are able to enumerate no more than ~1.7M prime numbers.

How does this settle with cryptographic prime number generation?
A Mersenne prime is a prime number that can be written in the form $M_p = 2^n-1$, and they’re extremely rare finds. Of all the numbers between 0 and $2^{25,964,951}-1$ there are 1,622,441 that are prime, but only 42 are Mersenne primes.


The second sentence is wrong.

What they meant to say is that there are 1,622,441 numbers of the form they mentioned in the first sentence below $2^{25,964,951}-1$.

There are 1,622,441 primes below 25,964,951 and thus 1,622,441 numbers of the form $2^p-1$ (with prime $p$) below $2^{25,964,951}-1$. See Wolfram Alpha.

There are over $2^{500}$ primes with exactly $2^{512}$ bits, so there are plenty to choose from for our RSA keys.

Take a look at the Prime number theorem which tells you that there are approximately $\frac{n}{\ln(n)}$ primes below $n$.