Voyager was headed out anyway. The plate was just deadheading. The chances being slim doesn’t mean they weren’t taking their best shot, just in case.
The exercise itself wasn’t worthless. I have no complaints.
Most mathematicians don’t really take set theory as foundational. In a formal sense it is, but most of us think of number and shape as primordial. It is hard to imagine a technology that doesn’t use numbers. If a civilization used base 30, we would probably catch on to that and then the sequence of primes is obvious.
It is also possible to use a variable base. We use one daily that can vary from 28 to 31 depending in a complicated way on higher digits and don’t give it much thought.
Yanno, you, well I, see and hear it said here and there that mathmatics is a subtle field, but I had to actually think about what you said for a second or two.
I’m not even sure it can be assumed that an extraterrestrial intelligence, assuming they reconstructed such a message into a grid would perceive a grouping of pixels as a representation of a physical object.
It was like a clue in a crossword puzzle. At first it made no sense at all, and then I looked at it again later and it was obvious.
When human minds work that way, speculating about alien minds is even more futile than speculating about the distant future.
In any first communication with aliens, the first, foremost, and most important piece of information you’re trying to convey is simply “we exist”. All the rest is just details. And even absent any other part of the message, the primes convey that important information.
The Pioneer plaques and Voyager records were guaranteed to be detected by sapient creatures, and their being detected by sapient creatures served an important purpose for the Voyager program as a whole. The creatures they were guaranteed to be detected by were us, and the purpose they served was to increase the “Wow, cool!” value of the missions, thereby increasing the likelihood of them receiving sufficient funding.
As for counting numbers, some mathematician once said “God created the counting numbers; all else was created by Man.”. But I would turn that around and say that what God created was the reals, and counting numbers are our creation. Yes, every culture we know of has developed them, but all of those are human cultures. So that tells us that humans are disposed towards discovering counting numbers. That doesn’t necessarily imply that aliens are. And in, for instance, Euclidean mathematics, it’s very easy to introduce the concept of real numbers, but you have to jump through a few extra hoops to get integers (and Euclid did jump through those hoops, and came up with some quite nice results about integers, but then, Euclid was after all a human).
That said, even if prime numbers aren’t as universally recognized as some folks assume, I’m not saying I have any better suggestions. You could also use, for instance, a diagram of the Pythagorean theorem… but if you’re sending anything other than a physical plaque, how do you encode that diagram, in a way that doesn’t depend on integers?
So, Chronos…it was just a marketing program 
{I agree that it was}
That’s an amazing comparison.
There’s another - also in daily use - that uses the much-maligned Base 60.
Thank you for opening my mind to seeing these things in new ways.
You could argue that that’s the same system, just in different decimal places. It starts off in base 60 for two places, then base 24 for one place (or arguably, one place of base 12 and one of base 2), then base 28-or-29-or-30-or-31 for one place, then base 12 for one place. And above or below those places, it’s all base 10. And the base 12 part of the system uses different names for all of the digits than all of the others, including some digits that are based on the usual digits but offset by two.
Quite right. But the point is that we use this very complex system every day and don’t give it a second thought (mostly, anyway).
Why ? I think the idea is that if we find a signal, just by the energy in the frequency band, and some sort of shape to the signal guides us to demodulate it, but then we can only recover a stream of numbers out of the signal, how can we tell between naturally created signal and a signal sent by intelligent life forms ?
The only logic to it is that if we could detect a higher than chance incidence of prime numbers in the stream, then it must be intelligently created; but if we don’t its not ruling out that it was intelligently created.
However, sending the 34.343243 form of numbers could maybe show the base they use, as repeating expansions ( like 0.343434343434 …34 repeating, which is 34/99 …) depends on the base.
1/3 is 0.3333333333, 3 repeating in decimal, base 10, but not in base 3, its only going to take one significant digit in base 3.
So we do have the situation that if they are sending information , such as common chat, books, or something, they might be sending either fractions, which show higher incidence of primes, OR long form expansions , which show repeating decimals for some fractions and not others, related to the base employed.
Still, what if the data is encrypted ? well the idea may be that the key exchange step can be identified by primes ?
If you saw the prime numbers in sequence in a stream, it is virtually impossible that it isn’t intelligence. We know of no direct formula that gets you the sequence of primes, you have to mine them from the sequence of integers. So there is essentially no reasonable physical process that could create them (unlike say, the Fibonacci sequence.)
A message that wants to draw attention to itself would probably look something like this:
Coding would be the simplest possible. Mark/space aka CW modulation. No numeric coding, just counting. It doesn’t get much simpler. In some ways this is a poor coding, but we want trivial recognition.
_x_xx_xxx_xxxxx_xxxxxxx_xxxxxxxxxxx_xxxxxxxxxxxxx_xxxxxxxxxxxxxxxxx_xxxxxxxxxxxxxxxxxxx
No issues with bases. That can come later. Just the above sequence has a vanishingly small chance of being random. You can extend it and repeat it, and increase the odds as much as you feel useful.
Once you have sent this for a while you can step up to more complex coding. Worrying about base of representation is not very important. Base 2 would be very hard to beat.
But for a message that isn’t intended to be a beacon to other lifeforms, and rather is intended for others of the same mob, current progress in coding here would suggest that the message would be close to impossible to even recognise, let alone decode. Techniques like spread spectrum, or time division, all use psuedo-random sequences to either frequency hop of choose a time division. You need to know the key to the sequence, and this isn’t trivially found. The entire point is that it behaves randomly, not necessarily to provide a form of encryption, but to avoid interference - both in causing and being affected by. The most efficient coding within the channel is also as close to noise as possible - using an efficient compression. So even though it isn’t intentionally encrypted, the end effect is not a great deal different.
If it was encrypted, the keys are not exchanged in the clear, but either known by each side in advance, or a session key is exchanged in an even more securely encrypted channel where the keys are known, or the channel is not interceptable without leaving a trace.
Given the speed of light is a hard barrier for speed of communications, unless we have a very very long lived ET, we can probably not expect to intercept general chit chat.
Compressed data also looks very much like random data. In fact, as the compression efficiency improves, the compressed data stream approaches complete indistinguishability from random data. Now, in practice, that’s not usually desirable, and usually even in compressed data we add a little bit of extra structured (i.e., distinguishable from random) data for things like checksums or error-correcting codes. There are some obvious ways to do that, and it wouldn’t be a huge surprise if aliens use some of the same methods, so we might be able to recognize that, if we had the complete uncorrupted data stream… but for an intercepted transmission, that’s a big if.
All that said, if your intent is to deliberately announce your presence to an alien race that might or might not know anything you know, you’re not going to be using compression or fancy encodings or anything like that. You’re going to use something super simple, like @Francis_Vaughan posted. It’s not very efficient, but it’s a short enough method that it really doesn’t need to be.
FWIW, many of the ideas of this thread, including the facts that primes are universal and that aliens could incorporate primes into a data stream sent to Earth to make it obvious how to interpret the data, are major plot points in Sagan’s novel “Contact.” These ideas even make it into the movie version.
But that’s like saying that every culture developed spears or bows. Sure, those are pretty universal inventions that, in different iterations, sprang up across the world unrelatedly.
But a bow or spear is designed to take what we’ve got (hands and arms with a certain range of motion) and to use that to strike at greater range.
The benefit (and selection pressure) is obvious. You can do more damage to an animal with a spear or arrow than with your bare hands, and you can strike without being struck back.
So given similar selection pressures (human biomechanics, the desire to kill animals to eat meat, the hope to do it safely without being hurt in return) it makes sense that bows and spears spring up.
Would aliens develop spears and bows? Maybe, maybe not. It depends on how their limbs are laid out, the other wildlife on their planet, what they like to eat…
While true to some extent, I would argue that without mobility (perambulation) and a large degree of dexterity, alien civilizations could not develop superior intellect or have the capability to build rocket ships.
Any alien life forms that arose from their version of primordial ooze to radio technological beings must surly have traversed a similar path as us. I always laugh when I see clunky aliens in movies. Do these look like hands that could build a microchip?
Are most microchips assembled by hand? 
Just to be pedantic - Pioneer 10 & 11 had the plaque with the naked images/menu selections on it. The Voyagers had Golden Records, with instructions on the back to build a record player.
Eh, any of the apes or monkeys has sufficient manual dexterity to make and use tools, but humans are the only ones of the lot who are any good at throwing things. Now, maybe throwing things was a necessary ability to be able to get where we got, but that’s because, without throwing weapons, there are still a number of predators who would be a serious threat to us.
But now imagine that, instead of the primates, it had been elephants who first developed sapience on our planet. Elephants don’t need any help protecting themselves against predators. Or maybe suppose that it was the octopuses: They still have relevant predators, but projectile weapons are near-useless underwater, so they’d need to develop some completely different sort of tool to protect themselves.
And that’s just among the creatures we know of on this planet. Who knows how much variety there might be among aliens, and what unique circumstances might lead them to develop or not develop various sorts of tools or concepts?