There are only finitely many \(D(4)\)-quintuples
A \(D(n)\)-\(m\)-tuple is an \(m\)-tuple \((a_1,a_2,\dots,a_m)\in\mathbb Z^m\) such that \(a_ia_j+n=\square\) for all \(1\leq i < j\leq n\). The \(D(1)\)-\(m\)-tuples which are better known as Diophantine \(m\)-tuples are well studied and Dujella showed that there do not exist Diophantine sex-tuples and at most finitely many quintuples [\textit{A. Dujella}, J. Reine Angew. Math. 566, 183--214 (2004; Zbl 1037.11019)]. In this paper the author finds the explicit bound \[ \max\{a,b,c,d,e\}<10^{10^{28}} \] for a \(D(4)\)-quintuple \(\{a,b,c,d,e\}\). This result goes along the same lines as Fujita's bound for the number of \(D(1)\)-quintuples [\textit{Y. Fujita}, Glasn. Mat., III. Ser. 45, No. 1, 15--29 (2010; Zbl 1211.11040)]. In particular, the result is achieved by considering simultaneous Pell equations.
scientific article; zbMATH DE number 5996727
scientific article