Do you mean why can't a square be twice another square?

Any integer can be broken down into a power of 2 times its 'odd part'. For example, 60 = 2² x 15, 1000 = 2³ x 125, 37 =2⁰ x 37, 38 = 2¹ x 19, etc. Thus if a square was twice another square, say m² = 2xn² let's think about the odd part of the numbers on each side of the equation. They must be the same and we can cancel them leaving just powers of 2 on both sides of the equation. Now comes a little bit of a subtle part. If we square an even number when we look at the result and break it into its odd part and powers of 2 the number of powers of two must be even; for example if we square 60 then the result is 2⁴ x 15², if we square 37 we end up with 2⁰ x 37², if we square 1000 we end up with 2⁶ x 125², etc. Thus if we had m² = 2 x n² the number of powers of 2 in m² is even, likewise the number of powers of 2 in n² is even but then we would have an even number of powers of 2 on the left hand side of the equation but (because of the extra 2) an odd number of powers of 2 on the right hand side of the equation and this can't be so as they're equal. Thus a square can't be twice another square

Hope this helps,

Penny