There is a test for showing that an integer is not a square.
Theorem: The square of an integer must be either a multiple of 4
. . . . . . . .or one more than a multiple of 4.
Proof: an integer must be even (2n) or odd (2n + 1).
If it is even, its square is: .(2n)² .= .4n²,
. . which is a multiple of 4.
If it is odd, its square is: .(2n + 1)² .= .4n² + 4n + 1 .= .4(n² + n) + 1
. . which is one more than a multiple of 4.
In other words, a square must end in 0, 1, 4, 5, 6, or 9.