The diophantine equation \(ax+by=c\) in \({\mathbb{Q}}(\sqrt{5})\) and other number fields (Q797621)

The diophantine equation (*) \(ax+by=c\), \((a,b)=1\), a,b,c integers in \({\mathfrak R}(\sqrt{5})\) is solved in integers \(\in {\mathfrak R}(\sqrt{5})\). The algorithm starts with a continued fraction development, called \(\lambda\)-fractions, of the form \(r_ 0\lambda +\frac{\epsilon_ 1}{r_ 1\lambda +}\frac{\epsilon_ 2}{r_ 2\lambda +}...\) where \(r_ i\in {\mathbb{Z}}^+\), \(r_ 0\in {\mathbb{Z}}\), \(\epsilon_ i=\pm 1\). Using a result of Leutbecher that every rational element in \({\mathfrak R}(\sqrt{5})\) has finite \(\lambda\)-fraction development, the algorithm proceeds by developing a/b as a unique finite \(\lambda\)-fraction \(P_ n/Q_ n\), using a nearest integer algorithm and then using the penultimate convergent \(P_{n-1}/Q_{n-1}\) to obtain the general solution to (*). The method mimics the one for the case (*) where a,b,\(c\in {\mathbb{Z}}\). The situation for other number fields is also discussed.