I suspect that the solution will always be integral for any set of equationsP + Q*R*S = W,
Q + R*S*P = X,
R + P*Q*S = Y,
S + P*Q*R = Z
where W, X, Y are Z
Can anybody prove or disprove this assertion?
My thinking (which does not constitute a proof) is that if P = a/b (relatively prime), and Q = c/d (relatively prime), and R and S are integers, then
1) R*S cannot be a multiple of d, or otherwise
(a/b) + (c/d)*R*S is not integral
2) R*S cannot be a multiple of b, or otherwise
(c/d) + (a/b)*R*S is not integral
3) S*a*c must be a multiple of b*d
4) R*a*c must be a multiple of b*d
Offhand, this feels inconsistent.