Rational sequences on algebraic curves

The thesis consists of seven chapters. In the first three chapters, the fundamental notions and some important theorems are given concerning algebraic and elliptic curves. Let E be an elliptic curve over Q described by y^2 =x^3+Kx+L where K,L∈Q. A set of rational points (xi,yi) ∈ E(Q) for i = 1,2,...,k, is said to be a sequence of consecutive cubes on E if the x-coordinates xi's of these points for i = 1, 2,... form consecutive cubes. In the fourth chapter of the thesis, we show the existence of an infinite family of elliptic curves containing a length-5-term sequence of consecutive cubes. Morever, it has been proved that these five rational points in E(Q) are linearly independent and hence the rank r of E(Q) is at least 5. In the fifth chapter of the thesis, given a set S of elements in a number field F, we discuss the existence of planar algebraic curves over F which possess rational points whose x-coordinates are exactly the elements of S. If the size |S| of S is either 4,5, or 6, we exhibit infinite families of (twisted) Edwards curves and (general) Huff curves for which the elements of S are realized as the x-coordinates of rational points on these curves. This generalizes earlier work on progressions of certain types on some algebraic curves. A sequence of rational points on an algebraic planar curve is said to form an r- geometric progression sequence if either the abscissae or the ordinates of these points form a geometric progression sequence with ratio r. In the sixth chapter of the thesis, we prove the existence of infinitely many rational numbers r such that for each r there exist infinitely many r-geometric progression sequences on the unit circle x^2+y^2 =1 of length at least 3.