We have
ab = cd
or \frac{a}{c}= \frac{d}{b} = \frac{m}{n} where m and n are in lowest terms or gcd(m,n) = 1
so an = cm and dn = bm
now n^2(a^2+b^2+ c^2+d^2)
= (na)^2 + (nb)^2 + (nc)^2 + (nd)^2
= (cm)^2 + (bn)^2 +(nc)^2 + (nd)^2= (b^2+c^2)(m^2 + n^2)
or a^2+b^2+c^2+d^2 = \frac{b^2+c^2}{n^2} (m^2+n^2)
as n is less than b^2+c^2 so it is product of 2 numbers > 1 so composite
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