I suspect that the solution will always be integral for any set of equations

**P + Q*R*S = W,** and:

**Q + R*S*P = X,** and:

**R + P*Q*S = Y,** and:

**S + P*Q*R = Z**

where

** W, X, Y are**** Z **are integral.

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.

Help, anybody?