Abstract
We provide several criteria to show over which quadratic number fields ℚ(√D) there is a nonconstant arithmetic progression of five squares. This is carried out by translating the problem to the determination of when some genus five curves CD defined over ℚ have rational points, and then by using a Mordell-Weil sieve argument. Using an elliptic curve Chabauty-like method, we prove that, up to equivalence, the only nonconstant arithmetic progression of five squares over ℚ(√409) is 72, 132, 172, 409, 232. Furthermore, we provide an algorithm for constructing all the nonconstant arithmetic progressions of five squares over all quadratic fields. Finally, we state several problems and conjectures related to this problem. © European Mathematical Society.
Original language | English |
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Pages (from-to) | 1211-1238 |
Journal | Revista Matematica Iberoamericana |
Volume | 29 |
DOIs | |
Publication status | Published - 1 Dec 2013 |
Keywords
- Arithmetic progressions
- Elliptic curve Chabauty method
- Mordell-Weil sieve
- Quadratic fields
- Squares