Mostly unrelated but this made me check to see if the Prime Number Shitting Bear is still online. It is! https://alpha61.com/primenumbershittingbear/
A mathematician, physicist, and engineer are taking a math test. One question asks "Are all odd numbers prime?"
The mathematician thinks, "3 is prime, 5 is prime, 7 is prime, 9 is not prime -- nope, not all odd numbers are prime."
The physicist thinks, "3 is prime, 5 is prime, 7 is prime, 9 ... experimental error, 11 is prime, 13 is prime, yes, they're all prime."
The engineer thinks, "3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, ..."
It'd be cool, if there was a Maker, that they make it so that some very large prime number is a message to all intelligent life capable of computing and deciphering it. This kind of message would penetrate even nested, simulated universes, and could only come from World 0, assuming the properties of numbers are independent of physical constants. Why not have a SETI-in-the-primes project, after all?
Every possible message is already there. Just like how Borges’ Library of Babel contains every possible book (within its limits of size and format)
In fact it seems like there must be many more non-primes than primes over any finite interval (for the simple reason that multiplication gets a lot of whacks at the piñata, e.g. producing a product landing in that finite interval.)
So to answer your implied question ("why bother commucating with primes anwyay?") I guess the value of limiting your message to "a really big prime number" is to make discovery hard, but still tenable. It should be pretty big, like big enough for a good bible sized book, say 1 gigabyte, and start with something really obviously artificial, or unexpected looking, like starting as a 1 with 20,000 zeroes, or maybe better, 20k perfect 010101 and then after a few thousand, 00110011, and from there establish a language protocol, and write the book. But you'd have to expect people to check only primes for these simple patterns, because it's really easy to produce meaningful sequences with addition, and really hard with multiplication (AFAIK).
Any finite message (containing a finite number of symbols drawn from a finite alphabet) can be represented as a positive integer.
Any positive integer n can be encoded in a prime, even a composite integer.
Use the following encoding system: write 1 n times, followed by a zero, followed by an arbitrary bit string.
For example, for 4: 11110xxxxxxxxxxxxx...
Using that encoding system, there will be an infinite number of primes that encode 4 - all the primes which start with 11110 in binary.
Thus, the set of all primes contains an infinite number of copies of every possible finite message.
The only reason to limit ourselves to primes is because its trivial to produce a meaningful composite, and very difficult to produce a meaningful prime. Take this message; if I was to associate it with a number (say by joining its characters as 7-bit ASCII), my money is that it's composite. (Not sure what the odds are, but I'd take 100:1 odds).
EDIT: period is `01110`, which is even, so I win.
Can we estimate how many gigabyte sized primes exist? I think one can do that using the prime number theorem. Let’s call that N. What is the probability one of those N messages is meaningful? I think it is a lot higher than you think it is. I think N is a very big number, even though it is very small compared to the number of composite gigabyte-sized numbers, it is still unimaginably large in absolute terms.
I mean, it's not that many if your expecting a message from God (I say half-jokingly).
Much more interesting is to discover a new communication, a message encoded in a very large number that is distinguished by being prime. (What message would you send? It would be interesting if it ultimately described some sort of recursive, reactive automata, the source code for a kind of learning, communicating intelligence. A gift, then. An awesome, terrible gift.)
You're welcome to look around for messages here: https://libraryofbabel.info/
I don't know if anyone has ever found one, though.
Sure, it can run alongside SETI-in-the-burn-marks-on-toast and SETI-the-flour-spilled-on-the-table projects.
Surely the engineer would round it to the nearest power of ten and thus discount all of the numbers?