Prime Numbers

10 Chapter 1 PRIMES!
naive subroutines, has bit complexity O(ln 3 z) for positive integer operands
x, y, z of comparable size, and so is polynomial-time. Similarly, taking a
greatest common divisor (gcd) is polynomial-time, and so on.
1.1.5 How primes are distributed
In 1737, L. Euler achieved a new proof that there are infinitely many primes:
He showed that the sum of the reciprocals of the primes is a divergent sum,
and so must contain infinitely many terms (see Exercise 1.20).
In the mid-19th century, P. Chebyshev proved the following theorem, thus
establishing the true order of magnitude for the prime-counting function.
Theorem 1.1.3 (Chebyshev). There are positive numbers A, B such that
for all x ≥ 3,
Ax
Bx