Hello, anncar!
A variation of Dan's explanation . . .
Prove that the sum of two odd squares cannot be a square.
If is even, , its square is: .
. . The square of an even number is a multiple of 4. .[1]
If is odd, , its square is: .
. . The square of an odd number is one more than a multiple of 4. .[2]
These are the only two forms for the square of an integer.
Let the odd integers be: .
The sum of their squares is: .
. .
Hence, is two more than a multiple of 4.
Since cannot be either of the forms [1] or [2], cannot be a square.