The length of the smallest co-prime chain between any two integers
The conjecture $\ell \leq 2$ seems to be open. It is stated as Conjecture 3 in a paper of Dowe. Dowe does show that the Goldbach conjecture implies that $\ell \leq 3$, and mentions that Alan Woods showed in his PhD thesis that $\ell$ is bounded. Perhaps the recent proof of the odd Goldbach conjecture also implies a definite and small bound.

If $\ell &gt; 1$ then $b-a$ is an Erdős-Woods number. In particular, if $\ell &gt; 2$ then both $b-a$ and $b-a-1$ have to be Erdős-Woods numbers. A059756 lists the first few Erdős-Woods numbers. While all numbers on that list are even, there are also odd Erdős-Woods numbers, some of which are listed in A111042. The smallest values of $a$ corresponding to given differences are listed in A059757. While this list is not increasing, it does seem as if the numbers are growing pretty fast. Any counterexample to $\ell \leq 2$ would to be such that both $b-a$ and $b-a-1$ are Erdős-Woods numbers, and in particular $b-a &gt; 430$ (per A059756). This suggests (but does not prove) that any such counterexample will be huge, and so for all practical purposes we can assume that $\ell \leq 2$.