# Prime Numbers (6 Answers To Common Questions)

Prime numbers are a central topic in number theory.  They are useful in cryptography, and they have many interesting properties as well.

Of course, there are some common questions about prime numbers to answer.

In this article, we’ll talk a little bit about prime numbers.  We’ll also answer some common questions about prime numbers as they relate to perfect squares, square roots, and even numbers.

Let’s get started.

## Common Questions About Prime Numbers

It is natural to want to connect prime numbers to other topics in number theory, such as perfect squares or even numbers.  However, let’s start with a simple one: the question of whether 1 is a prime number or not.

### Is 1 A Prime Number?

The number 1 is not a prime number.  There are several possible ways to see this. The number 1 is neither prime nor composite, since it has only 1 factor: itself.

First, a prime number p has two distinct positive factors: 1 and p.  The number 1 has only one positive factor: 1.

Second, we want to be able to write unique prime factorizations for any integer.  If 1 is a prime number, then we can write

• 6 = 1*2*3 and
• 6 = 1*1*2*3

as different prime factorizations of 6.  In fact, there would be infinitely many distinct prime factorizations of 6 (or of any positive integer for that matter).

Third, the number 1 is a perfect square, since 1 = 12.  However, a prime number cannot be a perfect square, as we prove below.

### Can A Prime Number Be A Perfect Square?

A prime number cannot be a perfect square.  To see this, let’s look at the definitions for these two terms:

• A prime number is a positive integer with only two different factors: 1 and itself (for example, 2, 3, 5, 7, 11, …).  So, for a prime number p > 1, the factors are 1 and p.
• A perfect square is a positive integer that can be written as a positive integer squared (for example, 1, 4, 9, 16, 25, …).  So, for a perfect square S, we can write it as S = n2, where n is a positive integer.

There is no overlap between the set of prime numbers and the set of perfect squares.  We’ll prove this now.

First of all, remember that 1 is a perfect square, but it is not a prime number.  So, the rest of the perfect squares are greater than 1.

For any perfect square S that is greater than 1, we can write it as S = n2, where n > 1.  This means that S will have at least 3 distinct factors: 1, n, and n2.

Since n is not equal to 1, n and n2 are not the same number.  Thus, any perfect square (except 1) has at least 3 distinct factors, while a prime number has only 2 distinct factors.

This means that any perfect square (except 1) is a composite number (it has factors other than 1 and itself).

### Do Prime Numbers Have Square Roots?

Prime numbers have square roots, but these square roots are always irrational.  Let’s prove this now.

We will use a proof by contradiction: assume that a prime number has a rational square root, and then prove that this is impossible.

If a prime number p has a rational square root, then

• p1/2 = a / b

where a and b are positive integers with no common factors (If a and b had a common factor, we could simplify the fraction and cancel until they had no common factors.  For example, we could reduce 5/15 to 1/3).

We can square both sides of the equation to get:

• (p1/2)2 = (a / b)2  [square both sides]
• p = a2 / b2  [simplify exponents]
• pb2 = a2  [multiply by b2 to clear denominator]

This equation tells us that p divides a2 (p is a factor of a2).  This means that p must divide a (p is a factor of a).

So, we can write a = pc, where c is a positive integer.  Substituting a = pc into our equation gives us:

• pb2 = (pc)2  [substitute a = pc]
• pb2 = p2c2
• b2 = pc2  [divide both sides by p]

This equation tells us that p divides b2 (p is a factor of b2).  This means that p must divide b (p is a factor of b).

So, a and b both have p as a factor.  However, this is a contradiction, since we assumed earlier that a and b had no factors in common!

Thus, p cannot have a rational square root.

So, the square root of a prime number p is irrational (it must involve a radical in simplified form).

### Can A Prime Number Be Even?

A prime number can be even in one special case: the number 2.

• This number is even, since it can be written as 2 = 2*1 (an even number can be written as 2 times an integer).
• This number is prime, since its two distinct factors are 1 and 2 (2 = 1*2).

Let’s prove that there are no other even primes.  Again, we’ll use proof by contradiction.

Assume that there is an even prime number p > 2.

This number p has at least the following factors:

• 1 (since 1 is a factor of every integer)
• 2 (since an even number always has a factor of 2)
• p (since the factors of a prime number p are 1 and p)

Note that 2 and p are different, since we said p > 2.  So p has at least 3 distinct factors (1, 2, and p).

This is a contradiction, since p is prime, which means p has only 2 distinct factors.

Thus, there is no even prime greater than 2.

### Can A Prime Number Be Divisible By 3?

A prime number can be divisible by 3, but only in one special case: the number 3.

• This number is divisible by 3, since it can be written as 3 = 3*1 (a number divisible by 3 can be written as 3 times an integer).
• This number is prime, since its two distinct factors are 1 and 3 (3 = 1*3).

Let’s prove that there are no other primes that are divisible by 3.  Again, we’ll use proof by contradiction.

Assume that there is a prime number p > 3 that is divisible by 3.

This number p has at least the following factors:

• 1 (since 1 is a factor of every integer)
• 3 (since a number that is divisible by 3 always has a factor of 3)
• p (since the factors of a prime number p are 1 and p)

Note that 3 and p are different, since we said p > 3.  So p has at least 3 distinct factors (1, 3, and p).

This is a contradiction, since p is prime, which means p has only 2 distinct factors.

Thus, there is no prime greater than 3 that is divisible by 3.

### What Are Twin Prime Numbers?

A pair of twin prime numbers is a set of two prime numbers p and p + 2 (they are separated by 2 units).

Some examples of pairs of twin prime numbers include:

• 3 and 5 (p = 3)
• 5 and 7 (p = 5)
• 11 and 13 (p =11)
• 17 and 19 (p = 17)
• 29 and 31 (p = 29)

As p gets larger, the pairs of twin primes get further apart.  For example, the distance between p = 11 and p = 17 is 6, while the distance between p = 17 and p = 29 is 12.

## Conclusion

Now you know some answers to common questions about prime numbers.  You also know how to think through questions about the factors of numbers.

You might also want to read my article on how many prime numbers there are.