Rational Points on a Class of Superelliptic Curves

Almost perfect powers in arithmetic progression

“…Lemma 10 and the following result of Győry are consequences of Lemma 11. In fact, as in Sander [14], the contributions of Wiles, Ribet and others give the following result on a more general Fermat equation than (7.1). Proof.…”

Almost perfect powers in consecutive integers

“…By using the contributions of Wiles, Ribet and others the following result on a more generalised Fermat equation has been given in Sander [10] and in [12,Lemma 13]. The next result of Bennett [1] is based on the hypergeometric method.…”

Rational points on the superelliptic Erdös Selfridge curve of fifth degree

“…Introduction, By a remarkable result of Erdos and Selfridge [3] in 1975. the diophantine equationwith integers A'>2 and »7^2, has only the trivial solutions .v = -/(/ = 1 m), v = 0. This put an end to the old question whether the product of consecutive positive integers could ever be a perfect power; for a brief account of its history see [7].From the viewpoint of algebraic geometry (1) represents a so-called superelliptic curve, and it seems to be more natural to ask for rational solutions (x: V) instead of integer solutions. Rational points on elliptic curves are well understood, but for general k and m, their nice arithmetic properties fade away.…”

“…It follows from Faltings's proof [4] of Mordell's conjecture that, for fixed k > \,m > 1 and k + m> 6, equation (1) has at most finitely many rational solutions (cf. [7]). It was shown by the second author [7] that, for k^2 and 2^w<4, all rational points (.v; v) on the superelliptic curve (1) are the trivial ones with \ --j (j = 1 m) and v = 0, except for the case k -m = 2 where we have exactly those satisfyingwiih coprime integers c\ ^ ±c'2-The second author also made the following CONJECTURE.…”

“…with integers A'>2 and »7^2, has only the trivial solutions .v = -/(/ = 1 m), v = 0. This put an end to the old question whether the product of consecutive positive integers could ever be a perfect power; for a brief account of its history see [7].…”