OK, that makes a bit of sense. But I don't know what it means for two real quantities to be relatively prime. Primality is about factoring, right? And you can only factor integers. So to observe primality, you'd have to partition interval measurements into an integral number time-units.

Can two reals be relatively prime no matter how you partition them into integers?

The ratios of several smallish prime integral time steps would bear out.

You're looking at the sky and you see a bright and otherwise unexpected signal. Then you see it again, some time later. You write that time down as T_1. You don't know of any astronomical phenomena with period T_1, so you wonder whether this could be a weird rare event. Then you see it again, spaced by T_2 this time, and you start to wonder whether it could be a purposeful signal. Then T_3 etc. How could they be linked? They're getting farther and father apart, so maybe it is a sequence with some significance. Maybe it's the fibonacci sequence for example. So you try dividing all the intervals by T_1. And you get 1,1.503,2.496,3.500,5.506... huh, lots of nearly-half-integers in there, maybe T_1 was supposed to be 2 units actually?

> so maybe it is a sequence

At that point, you introduce the assumption that the interval is quantized. Up until then, you're dealing with an arbitrary period of time, which is a real, and can't be made into sequences. Once you have sequences you have integers, and so you can have primes. But you can't just switch from reals to countable numbers without explanation, and then say that two reals can be relatively prime.

So begin with a sequence of 1 1 1 1 1 1 1 1 1 1. That establishes a baseline cadence of minutes, or hours or whatever. A clock signal if you will.

Then switch to primes. I think anyone looking at that kind of message will figure it out eventually. It can't be mistaken for a natural occurence.

> So begin with a sequence

That is the trick that I was calling out. The intervals are reals, and cannot be made into a sequence. Beginning with a sequence is even worse; what is in evidence (for the aliens) is a set of real intervals. That's what the aliens have to begin with.

Now explain how they get to primes.

If the precision between the intervals is good, how can you not get to primes? Of course the intervals would be real, but there would be a very discernible pattern.

It would stare you in the face, or what am I missing?

I disagree. Sequences of reals are still sequences. So maybe hypothesis one has nothing to do with integers, maybe it's that the intervals are increasing geometrically. You're still going to take ratios and and start having suggestive numbers fall out. "can't just switch ... without explanation". The point is that this putative alien observer would be looking for the explanation. Maybe the first few explanations they come up with are the wrong ones, but they are trivial to check. It's not like it's a research project to reject the Fibonacci hypothesis.So you would still have the right answer on day one.