On the Equation $Y^2 = X^5 + k$

Abstract

We show that there are infinitely many nonisomorphic curves $Y^2 = X^5 + k$, $k \in {\mathbb Z}$}, possessing at least twelve finite points $k>0$, and at least six finite points for $k<$. We also determine all rational points on the curve $Y^2=X^5-7$.