One of the ways to broadcast evidence of Math + Computing power could be to calculate this constant to an absurd-enough precision and have SETI (or someone) send the digits out into space.
Edit: in the sense that, we know how to encode a large set of primes in a single transcendental number, and by the way, here it is.
0.20305070110130 (...)
And now this whole shebang looks much less impressive, doesn't it?
I'm not sure why you ask? The above number already showed off the transition between 1 and 2 digit primes. And 0s inside the primes should not be a problem.
That last part means you can simply adjust your precision to your desired constraints. If you want to unambiguously represent the primes up through all the four-digit primes, simply change your padding accordingly: 0.000200030005....
Obviously the Buenos Aires Constant in the article is a neater representation, and relies on neater math. But GP's point was that you can stuff anything at all into a transcendent number.
Also, if I want to transmit a large set of primes (the complete calculated constant), I'll have to /retransmit/. How do I do that unambiguously? Well, "SOS". Transmit the set of primes. Pause long enough. Retransmit them. Ad infinitum. Ensure the pause is very very precise too.
It would be fun to pause for 2.<precision> seconds between retransmits.
Things are observable in the universe because we can tell when they are there and /not/ there. Any homogenous continuum ought to be undetectable (e.g. relativistic frame of reference, so my non-physicist brain says).
Edit: singular/plural bugfix
Of course, the aliens listening might not have proven it yet.