In our search for intelligent life in the universe (good idea or otherwise), we need in some way to prove we, too are intelligent. Classic science fiction dictates that prime numbers should be used to signal our presence. Why? Because prime numbers are unique, extremely unlikely to be naturally occuring, and the sign of a mathematically literate race.
Another beacon to pulse into the cosmos would be a constant of some sort. A single, repeating number, encoded in some easy to read fashion, which again signifies that we are here and intelligent. The Golden Ratio, Phi, is often suggested as well as Pi.
Quite how you send the message would be tricky in of itself. A series of binary signals (i.e. on-off-on-off) can more easily be interpreted as a number than some coded radio pulse, which needs a decoder i.e. it would be oibvious if you saw a signal that pulse the pattern ** *** ***** ******* *********** ************* but not so obviously if you had to demodulate a complicated number like Pi.
The Pioneer and Voyager probes carreid a famous plaque depicting various information about the origins of the probes., including some rather 70s looking images of the human form (seriously, no pubic hair?). The long lines with markings, show the frequencies of 14 pulsars that can used to locate the Sun relative to the galactic centre. Famously, most scientists that are shown this schematic are unable to decode it!
It would be interesting to see what each person on Earth would draw to represent us. Of course, the plaque may be meaningless to an alien life-form who, despite being intelligent, doesn’t understand the concept of drawing.
A simple, pulsing signal remains the best, and most simplistic method for attracting attention to ourselves. The problem of course is that we see no such signal out there ourselves. Assuming there are other civilizations in our galaxy, why do they appear to be silent?
Perhaps they are so different to us, that the message is simply undetectable or unrecognisable to our perception. Of course they may not be present at all, which would lead many to ask some very deep questions. But if they are there, and they are reaching out then we are not seeing the message. Could it be that the message is too advanced or subtle to appear obvious? Perhaps we need to be more creative in the ways we might send a message ourselves. In coming up with new ideas, we may stumble across something similar that has been under our noses all along.