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 k2 and m2, this equation only has the solutionsx = –j (j = 1, ..., m), y = 0 by a remarkable result ofErds and Selfridge 9] in 1975. This put an end to the old questionof whether the product of consecutive positive integers couldever 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 x0 in the case k = 2 andm = 3. In 1857, Liouville 18] derived from Bertrand's postulatethat for general k2 and m2, there is no solution with x0 ifone of the factors on the right-hand side of (1) is prime. Byuse of the Thue–Siegel theorem, Erds and Siegel 10] provedin 1940 that (1) has only trivial solutions for all sufficientlylarge kk₀ and all m. This was closely related to Siegel's earlierresult 30] from 1929 that the superelliptic equation y^k=f(x) has at most finitely many integer solutions x, y under appropriateconditions on the polynomial f(x). The ineffectiveness of k₀was overcome by Baker's method 1] in 1969 (see also 2]). In 1955, Erds 8] managed to re-prove the result jointly obtainedwith Siegel by elementary methods. A refinement of Erds' ideasfinally led to the above-mentioned theorem as follows.