Sorry; I blew it at the end of my "proof." Kill the final two paragraphs please. Proof of "revised Lunnon" retracted. Can replace the killed part with all this: But we know G divides GCD(2*s, c, a+b). Hence G divides (m+n)*GCD(2*m*n, m*n-k*k, m*n+k*k). Hence G divides (m+n)*2*GCD(m*n, k*k). But Lunnon-constraint implies that GCD(m*n,k*k)=GCD(m*n,k)... I think. Hence G divides 2*(m+n)*GCD(k, m*n). We know G divides both a-b = (n-m)*(m*n-k*k) and c = (n+m)*(m*n-k*k) hence G divides 2*(m*n-k*k)*GCD(m,n). Conclude: G must be a multiple of GCD(m+n, m*n-k^2) and must divide 2*GCD( (m+n)*GCD(m*n,k), (m*n-k*k)*GCD(m,n) ). Hence G must divide 2*(m*n-k*k)*m*n and 2*(m+n)*m*n*k. Hence G must divide 2*m*n*GCD( m*n-k*k, (m+n)*k). G must divide a*n-b*m = m*n*(n-m)*(n+m). Hence G must divide m*n*(n+m)*GCD(n-m, 2*k). G must divide a*n+b*m = m*n*(n*n+2*k*k+m*m). consider this and c we find G must divide (m+n)*GCD( m*n*(n-m), m*n-k*k ). Well, I'm clearly going insane here. The question is, can we conclude from some combination of all that crap, that G must divide 2*m*n*k? If so, the revised Lunnon conjecture, is proven.