Fuchs Clemens, Hutle Christoph, Luca Florian
1University of Salzburg, Hellbrunner Str. 34/I, 5020 Salzburg, Austria.
2School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits, 2050 South Africa.
Res Number Theory. 2018;4(3):29. doi: 10.1007/s40993-018-0121-2. Epub 2018 Jun 25.
The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence or finiteness of Diophantine triples in such sequences. Whilst the case of binary recurrence sequences is almost completely solved, not much was known about recurrence sequences of larger order, except for very specialised generalisations of the Fibonacci sequence. Now, we will prove that any linear recurrence sequence with the Pisot property contains only finitely many Diophantine triples, whenever the order is large and a few more not very restrictive conditions are met.
对取值于线性递归序列的丢番图三元组的研究是一个可追溯到亚历山大的丢番图的问题的变体，该问题在过去已被大量研究。与往常一样，主要问题是关于此类序列中丢番图三元组的存在性或有限性。虽然二元递归序列的情况几乎已完全解决，但对于高阶递归序列却知之甚少，除了斐波那契序列非常特殊的推广形式。现在，我们将证明，只要阶数足够大且满足一些不太严格的附加条件，任何具有皮索特性质的线性递归序列仅包含有限多个丢番图三元组。
