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Primality Testing

Finding large primes.

Density Of Primes

The prime distribution function $\pi$ (n) specifies number of primes $\leq$ n.

Theorem 33.37 $\stackrel{lim}{n \rightarrow \infty} \frac{\pi(n)}{n /ln n} \;=\; 1$

$\pi(n) \;\approx\; \frac{n}{ln n}$

For example, $\pi(10^9)$ $\approx$ 48,254,942.

The probability that randomly-chosen n is prime $\approx\; \frac{1}{ln n}$ . Thus, try $\frac{lnn}{2}$ odd numbers near n to find a prime with high probability.

For example, 100-digit number
$ln 10^{100} \;\approx\; 230$ . Try 115 odd numbers near 10¹⁰⁰.

About 1/230 100-digit numbers are prime.

Break input message M into numerical blocks smaller than n.

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