Let $R$ be an integral domain. Given $a,bin R$, then a division chain for $(a,b)$ is a sequence where we take $r_{-1}=a$, $r_0=b$, and for each $n>0$ we take $r_n=r_{n-1}s_n+r_{n-2}$ for some $s_nin R$. We say that the division chain terminates if $r_n=0$, and it terminates at length $n$ when $r_ineq 0$ for $i<n$.

These concepts, of course, have a lot to do with Euclidean domains, etc…

I ran across the following very interesting fact, proved by Cooke and Weinberger in 1975. Let $K$ be a number field, and let $R=mathscr{O}_{K,S}$, the ring of $S$-integers, where $S$ contains all the infinite places of $K$. Assume (some appropriate version of) GRH. If the unit group of $R$ is infinite, and $aR+bR=R$, then there is a terminating division chain for the pair $(a,b)$ of length $5$. Under some additional restrictions, like $S$ has a non-infinite place, or $K$ has a real embedding, the number $5$ can be lowered to $4$ or $3$. Moreover, they give examples showing that these numbers are best possible in some cases.

I’m interested if the following appears anywhere in the literature: For each $n>5$, there exists an integral domain $R$ such that for any $a,bin R$ with $aR+bR=R$, there is a terminating division chain for the pair $(a,b)$ of length $n$; and there is some pair of comaximal elements in $R$ that doesn’t have a smaller terminating length. The example will have to be somewhat complicated, since it won’t be a ring of integers over a number field.