Describe n N when the number of positive divisors of n is i

Describe n N when the number of positive divisors of n is: i) p2 for p a prime; ii) 105.

Solution

There exist an infinite number of primes.

Proof. Suppose that there are a finite number of primes, say p1, p2, . . . , pn. Let N = p1p2 · · · pn + 1. By the fundamental theorem of arithmetic, N is divisible by some prime p. This prime p must be among the pi , since by assumption these are all the primes, but N is seen not to be divisible by any of the pi , contradiction.