Let C : Y 2 = a n X n + · · · + a 0 be a hyperelliptic curve with the a i rational integers, n ≥ 5, and the polynomial on the right-hand side irreducible. Let J be its Jacobian. We give a completely explicit upper bound for the integral points on the model C, provided we know at least one rational point on C and a MordellWeil basis for J ‫.)ޑ(‬ We also explain a powerful refinement of the Mordell-Weil sieve which, combined with the upper bound, is capable of determining all the integral points. Our method is illustrated by determining the integral points on the genus 2 hyperelliptic models Y .

Abstract. In this paper, we study the Diophantine equation x 2 + C = 2y n in positive integers x, y with gcd(x, y) = 1, where n ≥ 3 and C is a positive integer. If C ≡ 1 (mod 4) we give a very sharp bound for prime values of the exponent n; our main tool here is the result on existence of primitive divisors in Lehmer sequence due Bilu, Hanrot and Voutier. When C ≡ 1 (mod 4) we explain how the equation can be solved using the multi-Frey variant of the modular approach. We illustrate our approach by solving completely the equations x 2 + 17 a 1 = 2y n , x 2 + 5 a 1 13 a 2 = 2y n , and x 2 + 3 a 1 11 a 2 = 2y n .

Abstract. We investigate power values of sums of products of consecutive integers. We give general finiteness results, and also give all solutions when the number of terms in the sum considered is at most ten.