Why is a prime number not a square number?

Question

Nova Allen · Accepted Answer

A prime number has #2# factors and squares have an odd number of factors.

All square numbers have an odd number of factors. #1# has the factor #1# #4# has the factors #1,2,4# #9# has the factors #1,3,9# #16# has the factors #1,2,4,8,16# #x^2# has the factors #1, x, x^2# A prime number by definition has exactly #2# factors - #1# and itself. Therefore no prime number is a square and no square number is prime.

Everly Taylor · Answer

A square number is of the form #axxa#.

This means it has factors other than #1# and itself. OK, I admit: #1=1xx1# is square, but #1# is (by definition) excluded from the list of primes. This is because a prime has two different dividors (1 and itself) and 1 has only one.

Benjamin Altschul · Answer

undefined A prime number is not a square number because a prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number cannot be divided evenly by any other number except 1 and itself.

On the other hand, a square number is the product of an integer multiplied by itself. For example, 4 is a square number because it equals 2 times 2.

Since a prime number cannot be divided evenly by any integer other than 1 and itself, it cannot be expressed as the product of two equal integers (i.e., it cannot be squared). Therefore, prime numbers and square numbers are distinct categories of numbers, with prime numbers being those that are indivisible (except by 1 and itself) and square numbers being those that are the result of multiplying an integer by itself.