Question

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

Collected in the board: Number Theory

Steven Zheng posted 11 hours ago

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

Steven Zheng posted 11 hours ago

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