On a Problem of Hajdu and Tengely

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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.
Samir Siksek, Michael Stoll
Added 15 Aug 2010
Updated 15 Aug 2010
Type Conference
Year 2010
Where ANTS
Authors Samir Siksek, Michael Stoll
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