If $$a,b,c,d$$ are four integers such that $$m = (ab+cd)^2-\dfrac{1}{4}(a^2+b^2-c^2-d^2)^2$$ is an nonzero integer. Show that $$|m|$$ must be a composite number

Question

Uniteasy Admin · Accepted Answer

Using the difference of squares formula,
$$m = (ab+cd)^2-\dfrac{1}{4}(a^2+b^2-c^2-d^2)^2$$
$$=[(ab+cd)+\dfrac{1}{2}(a^2+b^2-c^2-d^2)][(ab+cd)-\dfrac{1}{2}(a^2+b^2-c^2-d^2)]$$
$$=\dfrac{1}{4}[(a+b)^2-(c-d)^2][(c+d)^2-(a-b)^2] $$
$$=\dfrac{1}{4}(-a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)$$
Since $$m$$ is a nonzero integer, $$m$$ must be divisible by $$4$$.
Therefore, $$|m|$$ must be a composite number

Question

Answer