Intro

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Challenge:  Prove the following

If a quadruple of four non-negative integers $A$, $B$, $C$, $D$ is primitive, i.e., $\gcd(A,B,C,D)=1$, satisfies $$2(A^2 + B^2 + C^2 + D^2) = (A+B+C+D)^2$$ then any two add up to a sum of two squares: $$A+B = p^2+q^2$$ for some integers $p$ and $q$.