Nonextensibility of the pair \(\{1,3\}\) to a Diophantine quintuple in \(\mathbb Z[\sqrt{-2}]\) (Q2869648)
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scientific article; zbMATH DE number 6242870
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Nonextensibility of the pair \(\{1,3\}\) to a Diophantine quintuple in \(\mathbb Z[\sqrt{-2}]\) |
scientific article; zbMATH DE number 6242870 |
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3 January 2014
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Diophantine tuple
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Diophantine quintuple
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number field
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math.NT
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Nonextensibility of the pair \(\{1,3\}\) to a Diophantine quintuple in \(\mathbb Z[\sqrt{-2}]\) (English)
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The authors prove that the set \(\{1,3\}\) cannot be extended to a Diphantine quintuple in the ring \({\mathbb Z}[\sqrt{-2}]\). This result makes complete an earlier theorem of the first author, concerning the rings \({\mathbb Z}[\sqrt{-d}]\), where \(d\) is a positive square-free integer. In their proofs the authors combine results of Dujella and Pethő concerning systems of Pell equations, a theorem of Bennett on simultaneous approximations of square roots of rationals and Baker's method.
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