The next six exercises establish some estimates for the size of π(x), the number of primes...

The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.

Use Exercise to show that

Exercise

The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.

Let p be a prime and let n be a positive integer. Show that p divides  exactly

times, where t = [logp 2n]Conclude that if pr divides , then pr ≤2n.