Following the question gcd simplification, I was trying to see if $$\gcd\left(a y(p-q)-x(a-p q),y(a-pq)+x(p-q)\right)$$ with $\gcd(x,y)=1$ can be simplified.
I tried to eliminate the variable $q$ and ended up with the expression $$\gcd\left(a y(p-q)-x(b-p q),y(a-pq)+x(p-q)\right) \\ =\gcd\left((p^2-a)(x^2-ay^2),y(a-pq)+x(p-q)\right)$$ which seems to be not correct. A substitution of $$p=-13,a=151,x=28148430803,y=2290686812,q=-11$$ gives different values for the original expression and its supposed simplification.
Where is the mistake here? Is a simplification really possible for the original expression?