On a Problem of Hajdu and Tengely

Abstract. We answer a question asked by Hajdu and Tengely: The only arithmetic progression in coprime integers of the form (a2 , b2 , c2 , d5 ) is (1, 1, 1, 1). For the proof, we first reduce the problem to that of determining the sets of rational points on three specific hyperelliptic curves of genus 4. A 2-cover descent computation shows that there are no rational points on two of these curves. We find generators for a subgroup of finite index of the Mordell-Weil group of the last curve. Applying Chabauty’s method and the Mordell-Weil sieve, we prove that the only rational points on this curve are the obvious ones.

• Type: Conference
• Year: 2010
• Where: ANTS
• Authors: Samir Siksek, Michael Stoll
• Keywords: Algorithms, ANTS 2010, Coprime Integers, Rational Points, Specific Hyperelliptic Curves