Search

Numerous papers have studied the problem of determining upper bounds for the number of integer points on elliptic curves of the form y 2 = x 3 − m x , and quartic curves of the form X 2 − d Y 4 = k. Bounds for the number of integer solutions to such quartic equations typically depend on both of the coefficients d , k. The purpose of this paper is to examine more closely how the number of integer points on such quartic curves seems to depend almost entirely on the number of prime factors of k. This is done by focusing in on the problem of bounding the number of squares in certain recurrence sequences. In particular, using some arithmetic on elliptic curves, it is proved that infinitely many such sequences can have four squares, but that a fifth square remains elusive after extensive computation, suggesting that an absolute bound for the number of squares is more than likely, and also suggesting that the above assertion regarding the dependence on the number of prime factors of k. [ABSTRACT FROM AUTHOR]

Bumby proved that the only positive integer solutions to the quartic Diophantine equation 3X4 - 2Y2 = 1 are (X, Y) = (1, 1),(3, 11). In this paper, we use Thue's hypergeometric method to prove that, for each integer m ≥ 1, the only positive integers solutions to the Diophantine equation (m2 + m + 1)X4 - (m2 + m)Y2 = 1 are (X,Y) = (1, 1),(2m + 1, 4m2 + 4m + 3). [ABSTRACT FROM AUTHOR]