3 problems

Let $A,B,C\in\mathbb{Z}$ satisfy $\gcd(A,B,C)=1$. For prime exponents $p,q,r$ with … a non-trivial primitive integer solution of $Ax^p+By^q+Cz^r=0$ is a solution with $xyz\ne0$ and…

Generalized Fermat finiteness conjecture. For every integer $n\geq 6$, the equation

Tijdeman–Zagier conjecture. If $X,Y,Z\geq 3$, then the bases $A$, $B$, and $C$ have a common factor, that is, $\gcd(A,B,C)>1$. This is a generalized Fermat-type assertion excluding…