mathematics - How Many Undefined Magic Constants are there?

Magic Square: An nxn square where every horizontal,vertical, and diagonal line all add up to the same number.

Magic Constant: The number which every line with a magic square adds up to

So most magic squares, have a magic constant. But some do not. A fairly trivial answer is 0x0. Where no numbers are present, therefore it cannot sum up into any number.

A less than trivial answer but not complicated answer is 2x2. No possible configuration of 1,2,3,4 within a 2x2 block can have all lines add up to the same number. If it were possible it would theoretically be 5

So besides 0, and 2 are there any other positive dimensions for a magic square that do not have a magic constant? Technically making all possible configurations invalid magic squares?

Mathematical Proof is Appreciated!

Answer

From Wikipedia (my emphasis):

Normal magic squares of all sizes except 2 × 2 (that is, where n = 2) can be constructed. The 1 × 1 magic square, with only one cell containing the number 1, is trivial.

For a mathematical proof of this, see the Nrich article here. The idea is to turn an n x n square into a diamond (an n x n square standing on one corner), fill in that diamond with the numbers 1 to n^2 in the natural way, and then fold everything into the original square. Some pictures to demonstrate (also taken from the Nrich article):