Finiteness results for \(F\)-Diophantine sets. (English) Zbl 1396.11056
Let \(F\in \mathbb Z[X,Y]\) be a bivariate polynomial, then the authors of the paper under review call a set \(A\subseteq \mathbb Z\) \((F,m)\)-Diophantine if \(F(a,b)\) is an \(m\)-th power for any \(a,b\in A\) with \(a\neq b\). In case that \(F(x,y)=xy+1\) and \(m=2\) this definition coincides with classical Diophantine tuples. Under some mild technical assumptions they show finiteness results for \((F,m)\)-Diophantine sets. In particular, they give a complete characterization of all pairs \((F,m)\) for which infinite \((F,m)\)-Diophantine sets exist.
Reviewer: Volker Ziegler (Salzburg)
MSC:
| 11D61 | Exponential Diophantine equations |
Software:
SageMathReferences:
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