z = G2 where G is some integer which be factored using the F.T.A. into a unique product of primes (this fact is not used in the proof).
zk = z(p2+q2 )= G2(p2+q2) = G2p2+G2q2
Now we can assign m=Gp and n=Gq and get:
n2+m2 = G2p2+G2q2 = zk [done]
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Example/sanity-check: say z=25, p=2, q=7 : k = p2+q2 = 22+72 = 53
zk = 25*53 = 1325
G=sqrt(z)=5 so assign m=5*2 = 10 and n=5*7 = 35
m2+n2 = 102 + 352 = 100 + 1225 = 1325 = zk (ok)