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Erschienen in:
2010 | OriginalPaper | Buchkapitel
On a Problem of Hajdu and Tengely
verfasst von : Samir Siksek, Michael Stoll
Erschienen in: Algorithmic Number Theory
Verlag: Springer Berlin Heidelberg
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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.