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That cannot be true. It does not contain the message '4'. (Or any of the non-primes.) In fact it seems like there must be many more non-primes than primes over

That cannot be true. It does not contain the message '4'. (Or any of the non-primes.) 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).

skissane 7y ago
> That cannot be true. It does not contain the message '4'. (Or any of the non-primes.) 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.
- javajosh 7y ago
  I meant that the entire message is "4", not just part of the message. Obviously we can pick a subset of digits of any large prime and find anything we want. However, the odds of a very large prime (like a gigabyte) looking intentionally ordered over its entire length seems vanishingly small. 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.
  - skissane 7y ago
    > However, the odds of a very large prime (like a gigabyte) looking intentionally ordered over its entire length seems vanishingly small. 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.
    - javajosh 7y ago
      PNT, interesting! If I'm reading it correctly you can estimate the number of N digit primes as 10^N/N. That's basically 10^N, and for N=9 (a billion), that means there's (very very roughly) a billion primes with a billion digits. I mean, it's not that many if your expecting a message from God (I say half-jokingly).
      - skissane 7y ago
        Suppose somewhere out there, there is an ISO of Bible software, and it just so happens that when expressed as an integer it turns out to be prime – proof that the Bible is God's word, or freak coincidence? I think most people would go with the second option. And, the same would be just as true if it was an ISO of some other religious scripture, such as the Quran or the Vedas.
        javajosh 7y ago
        Well it would be an impressive feat, regardless. But of course there are a lot of degrees-of-freedom if you were to attempt this. Tiny formatting and encoding decisions would all greatly influence the resulting integer. Because it's so sensitive to chosen input, I would say even if you managed to produce a prime, it's not very satisfying as a sort of Godly checksum and signature. 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.)

skissane 7y ago

> That cannot be true. It does not contain the message '4'. (Or any of the non-primes.) 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.

javajosh 7y ago
I meant that the entire message is "4", not just part of the message. Obviously we can pick a subset of digits of any large prime and find anything we want. However, the odds of a very large prime (like a gigabyte) looking intentionally ordered over its entire length seems vanishingly small. 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.
- skissane 7y ago
  > However, the odds of a very large prime (like a gigabyte) looking intentionally ordered over its entire length seems vanishingly small. 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.
  - javajosh 7y ago
    PNT, interesting! If I'm reading it correctly you can estimate the number of N digit primes as 10^N/N. That's basically 10^N, and for N=9 (a billion), that means there's (very very roughly) a billion primes with a billion digits. I mean, it's not that many if your expecting a message from God (I say half-jokingly).
    - skissane 7y ago
      Suppose somewhere out there, there is an ISO of Bible software, and it just so happens that when expressed as an integer it turns out to be prime – proof that the Bible is God's word, or freak coincidence? I think most people would go with the second option. And, the same would be just as true if it was an ISO of some other religious scripture, such as the Quran or the Vedas.
      - javajosh 7y ago
        Well it would be an impressive feat, regardless. But of course there are a lot of degrees-of-freedom if you were to attempt this. Tiny formatting and encoding decisions would all greatly influence the resulting integer. Because it's so sensitive to chosen input, I would say even if you managed to produce a prime, it's not very satisfying as a sort of Godly checksum and signature. 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.)