FAMILIES OF CUBIC ELLIPTIC CURVES CONTAINING SEQUENCES OF CONSECUTIVE POWERS

Let E be an elliptic curve over Q defined by y(2) = ax(3) + bx(2) + cx + d. We say a sequence of rational points (x(i), y(i) ) is an element of E (Q) , i = 0, 1, ..., l, forms a sequence of consecutive n-th powers on E of length l whenever the sequence of x-coordinates, x(i), i = 0, 1, ..., l, consists of consecutive powers of degree n in the form x(i) = (g + i)(n) On, for some rational g. Applying the known Mestre's theorem, for an arbitrary natural number n >= 2, we produce a one-parameter family of elliptic curves over Q which contains an 8-term sequence of consecutive n-th powers. Furthermore, we show that for n = 2 and 3 the associated families of elliptic curves are of generic rank >= 6 and 7, respectively. We also provide an explicit set of linearly independent points for those families. Finally, according to our limited trial conducted for 4 <= n <= 50, we discovered that the generic rank of the corresponding families is >= 7. We guess that this holds for all n >= 4; however, we are not able to prove it at this time.