Constructions of Diagonal Quartic and Sextic Surfaces With Infinitely Many Rational Points

In this paper, we construct several infinite families of diagonal quartic surfaces ax⁴ + by⁴ + cz⁴ + dw⁴ = 0 (where a, b, c, d are non-zero integers) with infinitely many rational points and satisfying the condition abcd is not a 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 ax⁶ + by⁶ + cz⁶ + dwⁱ = 0, i = 2, 3, or 6, with infinitely many rational points.