On a Problem of Hajdu and Tengely

  • Samir Siksek
  • Michael Stoll
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)


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.


Rational Point Galois Group Arithmetic Progression Hyperelliptic Curve Weierstrass Point 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Samir Siksek
    • 1
  • Michael Stoll
    • 2
  1. 1.Institute of MathematicsUniversity of WarwickCoventryUK
  2. 2.Mathematisches InstitutUniversität BayreuthBayreuthGermany

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