On a Problem of Hajdu and Tengely
We prove a result that finishes the study of primitive arithmetic progressions consisting of squares and fifth powers that was carried out by Hajdu and Tengely in a recent paper: The only arithmetic progression in coprime integers of the form (a 2, b 2, c 2, d 5) 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, we prove that the only rational points on this curve are the obvious ones.
KeywordsRational Point Galois Group Arithmetic Progression Hyperelliptic Curve Weierstrass Point
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