Why is a prime number not a square number?

2 Answers
MeneerNask
Dec 27, 2016

A square number is of the form #axxa#.

Explanation:

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.

EZ as pi
Oct 7, 2017

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

Explanation:

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.

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