One of the most commonly forgotten things about mathematics is that all statements are if and only if statements. This is one of the very first things that my number theory professor reminded us of.

By definition, a number is a prime power if it is a positive integer power of a prime number. This means that we take a prime number, raise it to some power, and the resulting number is a prime power. So if we let n equal some positive integer, then raising a prime number to n, will result in a prime power.

For example:

2 is a prime power, as 2^1 = 2

4 is a prime power, as 2^2 = 4

8 is a prime power, as 2^3 = 8

16 is a prime power, as 2^4 = 16

This side is easy to see. Let’s look at another statement that we can make about prime powers. If we let n denote some positive integer, then a number is a prime power only if we can raise that number to the power of 1/n and come to a prime number.

For example:

2^(1/1)= 2

4^(1/2)= 2

8^(1/3)= 2

16^(1/4) = 2

Let’s take a look at your claim that 1 should be included as a prime power. We can raise any prime number to 0 and arrive at 1.

2^0 = 1

How do we fulfill the only if statement? Let’s examine the number 16 for a minute. We saw that:

2^4 = 16

16^(1/4) = 2

If we follow the same logic structure, then we should be able to do the same when we raise 2, or any prime number (or any number) to a power. Let’s try out your statement that 1 should be included because we can raise any number to zero and get 1.

2^0 = 1

1^(1/0) = undefined

If your proposition was correct, we should have been able to raise 1^(1/0) and come back to 2. But we can’t. In fact, 1 raised to any power is just 1. And 1 is not a prime number.