In this note we construct several infinite families of diagonal quartic surfaces \begin{ equation*} ax^4+by^4+cz^4+dw^4=0, \end{ equation*} where $a,b,c,d\in\Z\setminus\{ 0\}$ with infinitely many rational points and satisfying the condition $abcd\neq \square$. In particular, we present an infinite family of diagonal quartic surfaces defined over $\Q$ with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type $ax^6+by^6+cz^6+dw^i=0$, $i=2$, $3$, or $6$, with infinitely many rational points.
Categories: math.NT