5.

(a) Use Theorem 2.53 to approximate the number of primes whose binary represen- tation has between 512 and 1024 bits.

(b) Assume that there are about 1077 atoms in our universe and that this universe is about 100 years old. Assume also that, from the beginning of time, every atom in the universe requires, each second, 10 many primes with b bits (where 512 < b1024) for its own mysterious cryptographic purposes. Assuming that these atoms never use a prime that they (or some other atom) have used before, would there still be such primes available today? If so, for how long?