Set Theory Show a proof by contradiction that there does not

(Set Theory)

Show a proof by contradiction that there does not exist ints x, y such that
x^2 = 8y + 3.

Solution

Here we assume that integers x and y exist that satisfy the given equation. Then,

x^2 = A positive number always (no matter x is positive or negative).

But 8y+3 may be negative also if y is a negative integer.

In that case, given equation is not true.

So our assumption is false in case of negative integers. Thus there does not exit ints x ,y such that x^2=8y+3

Proved.