7 problems

Consider the generalized Fermat equation … A solution $(a,b,c)\in\mathbb{Z}^3$ is primitive if $\operatorname{gcd}(a,b,c)=1$ and trivial if $abc=0$. The Beal Prize conjecture. If…

Let $p,q,r$ be positive integers, and call a solution $(\alpha,\beta,\gamma)$ of $x^p+y^q=z^r$ primitive when $\gcd(\alpha,\beta,\gamma)=1$. Fermat–Catalan's conjecture. For any tr…

Let $K$ be a number field, and let $A,B,C$ be non-zero elements of $\mathcal{O}_K$. Let $\Omega$ be the subgroup of roots of unity inside $\mathcal{O}_K^{\times}$. Suppose … for an…

The generalized Fermat–Catalan conjecture. The primitive solutions arising from the ten identities

The abc conjecture. As a consequence of the abc conjecture, this equation has only finitely many such solutions, except for $1^r+2^3=3^2$.

Laradji–Mignotte–Tzanakis conjecture. If $q$ is not of the form

Let $A,B,C\in\mathbb{Z}_{\neq 0}$ and let $q,r,p\in\mathbb{Z}_{>0}$. A primitive solution of … is an integer solution $(x,y,z)$ satisfying $\gcd(Ax,By,Cz)=1$, and the signature is…