I am thinking the answer is yes, and if I can show that for any pair of distinct primes , the set is dense in the reals, then let . Now, we define and for each , choose such that . Then, let . From there, I would know how to show that this is a GCD-Cauchy sequence whose limit is . But, this construction relies on that density property I mentioned. I have been trying to think of an argument for it (or against it), but so far I have yet to come up with anything substantial.