3 problems

Let $S$ be a finite set of primes. Consider integer solutions $(a,b,c,x,y,z,p,q,r)$ of the generalized Fermat equation $eqref{eq:genfermat}$ satisfying $\gcd(ax,by,cz)=1$, $p,q,r>0$…

Let $a,b,c$ be nonzero integers, and let $(x,y,z,p,q,r)$ be integers satisfying the generalized Fermat equation $eqref{eq:genfermat}$, with $\gcd(x,y,z)=1$, $p,q,r>0$, and…

Let $p,q$ and $r$ be positive integers satisfying … A solution $(X,Y,Z)$ is called primitive here when $XYZ\ne0$ and $\gcd(X,Y)=1$. Generalized Fermat conjecture. All primitive sol…