SEQUENCES OF CONSECUTIVE SQUARES ON QUARTIC ELLIPTIC CURVES

Let C : y(2) = ax(4) + bx(2) + c, be an elliptic curve defined over Q. A set of rational points (x(i), y(i)) is an element of C(Q), i = 1, 2, . . . , is said to be a sequence of consecutive squares if x(i) = (u + i)(2), i = 1, 2, . . . , for some u is an element of Q. Using ideas of Mestre, we construct infinitely many elliptic curves C with sequences of consecutive squares of length at least 6. It turns out that these 6 rational points are independent. We then strengthen this result by proving that for a fixed 6-term sequence of consecutive squares, there are infinitely many elliptic curves C with the latter sequence forming the x-coordinates of six rational points in C(Q).