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This is a case where the general problem is easier to reason about than a specific problem. Forget about prime numbers for a moment: it is impossible to achiev

This is a case where the general problem is easier to reason about than a specific problem. Forget about prime numbers for a moment: it is impossible to achieve lossless compression that always succeeds in compressing N bits into fewer than N bits. This is because 2^M is smaller than 2^N when M<N. You can't map X objects into fewer than X objects without losing information. So the issue is not that prime factorisation fails to reliably compress things. It's that every possible method you can try would fail to do so, no matter how advanced or fancy it is. Sometimes you'll manage to make the data smaller, sometimes you won't. Relatedly, there's an interesting proof that there exist infinite prime numbers that uses the idea of data compression in it: https://en.m.wikipedia.org/wiki/Euclid%27s_theorem#Proof_using_the_incompressibility_method https://en.m.wikipedia.org/wiki/Euclid%27s_theorem#Proof_usi...

meltyness 4y ago
I was and am aware of this result, but I find myself under the spell of Constructor Theory's claims that information theory has fundamental issues, and simply don't know enough to verify the apparent fact that this is impossible, theoretically. For example, as I read your argument, I observe that you've divorced representation from the situation -- that is for a given number that the number of digits of information and number of bits of information aren't fungible, and can only be approximated as a metric of the amount of information contained. That proof regaring infinitely many primes is clever though, thank you for that tidbid.
- FartyMcFarter 4y ago
  Here are two different ways of thinking about it which IMO are quite convincing: - If you had a reliable way to losslessly compress any N bits of information into N-1 bits of information, you could use this iteratively to compress N bits into N-1 bits, then into N-2 bits, then N-3 bits, and so on until you've compressed anything into 1 bit of information. This is obviously ridiculous, so the initial assumption that we have such compression can't be true. - Alternatively, try for yourself to make a function that maps 4 objects into 2 objects. Say mapping the input space "red, green, blue, or yellow" into the output space "white, or black". You won't be able to make this function reversible no matter what you do - it's easy to demonstrate by brute force as you can try all possible functions. If your input space is 2, your output space is at most 2, not 4. You can extend the same intuitive idea to prove that N bits of information can't always be compressed into N-1 bits by the same method.

meltyness 4y ago

I was and am aware of this result, but I find myself under the spell of Constructor Theory's claims that information theory has fundamental issues, and simply don't know enough to verify the apparent fact that this is impossible, theoretically. For example, as I read your argument, I observe that you've divorced representation from the situation -- that is for a given number that the number of digits of information and number of bits of information aren't fungible, and can only be approximated as a metric of the amount of information contained. That proof regaring infinitely many primes is clever though, thank you for that tidbid.

FartyMcFarter 4y ago
Here are two different ways of thinking about it which IMO are quite convincing: - If you had a reliable way to losslessly compress any N bits of information into N-1 bits of information, you could use this iteratively to compress N bits into N-1 bits, then into N-2 bits, then N-3 bits, and so on until you've compressed anything into 1 bit of information. This is obviously ridiculous, so the initial assumption that we have such compression can't be true. - Alternatively, try for yourself to make a function that maps 4 objects into 2 objects. Say mapping the input space "red, green, blue, or yellow" into the output space "white, or black". You won't be able to make this function reversible no matter what you do - it's easy to demonstrate by brute force as you can try all possible functions. If your input space is 2, your output space is at most 2, not 4. You can extend the same intuitive idea to prove that N bits of information can't always be compressed into N-1 bits by the same method.