In the paper the domain of definition of a 1 chain is extended to the field \(K (\sqrt{-3})\): Definitions. A sequence of integers \(\{u_ i\}\) in \(K (\sqrt{-3})\), with at least three terms, in which any three consecutive terms satisfy the relation \(u_{n-1}\cdot u_{n+1}=u_ n^ 3+1\), is called a 1 chain. A 1 chain that contains only rational integers is denoted a 1+chain if all elements are positive integers and a \(1\pm\)chain if it contains at least one negative integer. A 1 chain \(\dots,u_{n-1},u_ n,u_{n+1},\dots\) is written as \(\langle u_{n- 1},u_ n,u_{n+1} \rangle\). Results regarding such generalized 1 chains, similar to S. P. Mohanty’s for 1+chains [J. Number Theory 9, 153-159 (1977; Zbl 0349.10010)], are given as well as new results concerning the least element(s) of a 1 chain.
When permitting the elements of a 1 chain to be integers in \(K (\sqrt{- 3})\), the proofs require considerably more details than the corresponding theorems for 1 chains containing only real elements. E.g. there are infinitely many 1 chains \(\langle u,s,t \rangle\) that satisfy \(| u |=| s |\) and \(| t |>| s |^ 2\). In these 1 chains \(s\) and \(u\) are the least elements but \(| t |<| s |^ 2\) is not satisfied, as for all real 1 chains. A generalization, valid for all 1 chains, is proved:
If \(\langle u,s,t \rangle\) is a nonsingular 1 chain, \(u,s,t\) are integers in \(K (\sqrt{-3})\) and \(s\) is a unique least element i.e. \(| u |>| s |\) and \(| t |>| s |>1\), then \(| u |<| s |^ 2\) and \(| t |<| s |^ 2\). On the other hand, if \(1<| s |<| u |<| s |^ 2\), then \(| t | \geq | s |\), i.e. \(s\) is the least (but not necessarily unique) element. Further all solutions of the diophantine equation \(\alpha^ 3+\beta+1=\alpha \beta \Gamma\), where \(\alpha,\beta,\Gamma\) integers in \(K (\sqrt{-3})\), are given. In addition to the known particular and parametric solutions, a finite set of solutions exists in \(K (\sqrt{-3})\) as well as \((\alpha \sigma,\beta)\) and \((\hat\alpha,\hat\beta)\) where \(\sigma\) is a unit in \(K (\sqrt{-3})\) and \((\alpha,\beta)\) any other solution.
A table with all 1+chains in which the least element \(\leq 78567\) is appended and it gives just one instance: \(\langle 3002, 1209, 588665 \rangle\) and \(\langle 13205, 1209, 133826 \rangle\) where two nonidentical 1+chains have the same least element.