First, put the expression in a more familiar form : why can't ever be a perfect square ? For to be a perfect square, we need to have : for some integer , that is, . Thus, saying that no integer a satisfies this condition is equivalent to showing that is never divisible by three. The "alternative" question is in fact necessary to solve the first question, so see the following :why can't 5+3n ever be a perfect square? (n is an integer)
Suppose is divisible by three, so is divisible by three, so obviously is not divisible by three ( ).why isn't x^2-5 divisible by three? (x is an integer)
Suppose x is not divisible by three, so . Therefore, , that is, , or, pushing even further, . Note that squares can only be equal to or modulo 3, but we already considered the case when it is equal to 0 modulo 3 (it is divisible by three). So assume , so .
Conclusion : is either equal to 1 or 2 modulo 3, and thus cannot be divisible by three.
Final conclusion : cannot be a perfect square.
Does that make sense ?