# Prime number of divisors, where does one use this?

I recently asked this question Total number of divisors is a prime.

How useful is this property? What are its applications?

• As you can see from the answers to your first question, there aren't many numbers with this property, and they aren't extremely interesting. – Najib Idrissi Nov 5 '13 at 1:33
• So it has no applications currently whatsoever? – kintoki Nov 5 '13 at 1:37
• I wouldn't necessarily say that. It may be a useful little lemma for some proof. But, it's not that profound of a result. – mojambo Nov 5 '13 at 1:41
• Generally, this is a bad way to find number classes have have interesting applications. There are lots of classes of numbers, and very few of them have useful applications. – Thomas Andrews Nov 5 '13 at 2:32
• You might as well ask a monkey to pick a random assortment of mathematical adjectives until a well-defined property is described and then ask for the potential uses and applications. – anon Nov 5 '13 at 2:51

## 2 Answers

The property is as useful as to say that a given number $n$ is of the form $p^{q-1}$ for primes $p$ and $q$. Indeed, $\tau(n)$ is prime if and only if $n=p^{q-1}$. Of course, the arithmetic function $\tau (n)$ is important in number theory. Even more so the function $\sigma(n)$, the sum of all divisors. The Riemann hypothesis is equivalent to an elementary statement on $\sigma(n)$, see the paper of Lagarias (2002), "An elementary problem equivalent to the Riemann hypothesis".

I got this helpful answer at Ask Dr.Math so I'm posting it.

A number with just two divisors is prime, and primes are useful. But is it useful to characterize the numbers with exactly 17 divisors (for example)? I don't know.

The concept that has applications is that of "multiplicative function". In number theory, a function f is called multiplicative if whenever m and n are relatively prime, f (mn)= f (m) f (n). The function d (n) that counts the number of divisors of n is one example.

If f is multiplicative, f is completely determined once you know its values on prime powers. That's the key thing we need to know to characterize the numbers n for which d (n) is prime. So in my opinion, the importance of this problem is that it helps get you more familiar with the idea of a multiplicative function. It's this concept that has the applications.