Question

proof unknown "a" is congruent to 2 or negative 1 modulo 4, it mean "a" is not prefect square, such as a=b^2, "b" is an integer.

Answer 1

Assume that a=2mod(4) or a=-1mod(4).

To show that a is not perfect square.

Proof by contradiction,

Suppose if possible that a is perfect square

Thus a=b² for some integer b

Case 1) if b is even.

b=2k for some integer k

Thus, b²=4k²

That is, a=4k²

Hence 4|a

That is, a = 0 mod(4)

Which is contradiction to our assumption.

Case 2) if b is odd.

b=2k+1 for some integer k

b²=4k²+4k+1

That is, a=4k²+4k+1

a=4x+1 where x=k²+k

a=1 mod(4)

Again contradiction to our assumption.