Rational Points on a Class of Superelliptic Curves

A famous Diophantine equation is given by y k =( x +1)( x +2)…( x + m ). (1) For integers k ⩾2 and m ⩾2, this equation only has the solutions x = − j ( j = 1, …, m ), y = 0 by a remarkable result of Erdős and Selfridge [ 9 ] in 1975. This put an end to the old question of whether the product of consecutive positive integers could ever be a perfect power (except for the obviously trivial cases). In a letter to D. Bernoulli in 1724, Goldbach (see [ 7 , p. 679]) showed that (1) has no solution with x ⩾0 in the case k = 2 and m = 3. In 1857, Liouville [ 18 ] derived from Bertrand's postulate that for general k ⩾2 and m ⩾2, there is no solution with x ⩾0 if one of the factors on the right‐hand side of (1) is prime. By use of the Thue–Siegel theorem, Erdős and Siegel [ 10 ] proved in 1940 that (1) has only trivial solutions for all sufficiently large k ⩾ k 0 and all m . This was closely related to Siegel's earlier result [ 30 ] from 1929 that the superelliptic equation y k = f ( x ) has at most finitely many integer solutions x , y under appropriate conditions on the polynomial f ( x ). The ineffectiveness of k 0 was overcome by Baker's method [ 1 ] in 1969 (see also [ 2 ]). In 1955, Erdős [ 8 ] managed to re‐prove the result jointly obtained with Siegel by elementary methods. A refinement of Erdős' ideas finally led to the above‐mentioned theorem as follows.