Prime number distribution

Prime number function: \pi(n)=|\{p\in \mathbb{P}| p \leq n\}|

Prime number theorem:

\lim_{n \to \infty} \pi(n) \approx\frac{n}{\ln(n)}

This can be used to estimate the number the number of prime numbers.

ln(n) is the natural logarithm of n — it's the logarithm with base e (where e ≈ 2.71828, Euler's number).

l-bit guesstimation function: \begin{align} s(n) &= \frac{n}{\ln(n)} \\ t(l) &= s(2^{l})-s(2^{l-1}) \end{align}

Using the Prime Number Theorem to estimate 512-bit primes:

A 512-bit number is approximately in the range [2^{511}, 2^{512}].

Using π(n) ≈ n/ln(n), the number of primes up to 2^{512} is roughly:

\pi(2^{512}) \approx \frac{2^{512}}{\ln(2^{512})} = \frac{2^{512}}{512 \cdot \ln(2)} \approx \frac{2^{512}}{355}

This is approximately 2^{502} primes.

The number of primes between 2^{511} and 2^{512} is roughly:

\pi(2^{512}) - \pi(2^{511}) \approx \frac{2^{512}}{355} - \frac{2^{511}}{354} \approx 2^{501}

To compare with atoms in the universe:

• Atoms in the universe: \approx 10^{89} = 2^{296}
• 512-bit primes available: \approx 2^{501}

So there are approximately 2^{(501-296)} = 2^{205} times more 512-bit primes than atoms in the universe.

Why this matters for cryptography:

When RSA generates a 2048-bit key, it randomly selects from roughly 2^{2037} available primes. The probability of randomly guessing the same prime twice is about 1 in 2^{2037}—effectively impossible.