Attempt one of the following:

(i) Discuss pseudoprimes and primality testing. Find all bases for which 57 is a Fermat pseudoprime. Determine whether 57 is also an Euler pseudoprime for these bases.

(ii) Write a brief account of various methods for factoring large numbers. Use Fermat factorization to find the factors of 10033. Would Pollard's $p-1$ method also be practical in this instance?

(iii) Show that $\sum 1 / p_{n}$ is divergent, where $p_{n}$ denotes the $n$-th prime.

Write a brief account of basic properties of the Riemann zeta-function.

State the prime number theorem. Show that it implies that for all sufficiently large positive integers $n$ there is a prime $p$ satisfying $n<p \leqslant 2 n$.