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Comment by dmvaldman | Hacker News Reader

The reason why this problem is difficult is because primes have everything to do with multiplication (every number is the product of primes), and summing numbers have everything to do with...well, addition. To give additive properties of numbers in terms of primes is like putting a round peg in a square hole.

The reason why this problem is important, is that in the grand scheme of things, it'a more about the math and less about the applications. People have already assumed it to be true, and still not applied it to anything practical like cryptography, as far as I know. Please correct if not.

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> every number is the product of primes

Except prime numbers.

> People have already assumed it to be true, and still not applied it to anything practical like cryptography

How can you use something that you didn't prove?

Can you explain your question? It seems obvious that you can take something that is not proven as an axiom and try to use it in some practical applications. Maybe to disastrous results if it ends up being wrong, but no worse than what is already a risk with bugs in implementation.

> Maybe to disastrous results if it ends up being wrong

This is what I meant. Bugs are never introduced intentionally in this manner. Though I suppose we can say that most encryption is used based on an unproven assumption, so I see your and dmvaldman's point!

Also, Golbach's conjecture has been verified up from 5 to 4000000000000000000 computationally.

But in general, it's common for mathematicians to assume a conjecture is true and go on from there, because if they arrive at a contradiction, then they can conclude the conjecture false (or that another interesting fact may be logically equivalent to the conjecture).

Pure mathematics consists entirely of such assertions as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true…. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. - Bertrand Russell

Easily, act as if you proved it. We are generally very hesistent to do this (with good reason), but sometimes practicality takes over.

For example, we have not proven that are current construction of mathematics contains no contradictions (indeed, we have proved that such a proof is impossible), however, we continue to act as if it does.

We have also not proved that factoring numbers is hard, or that NP/=P, but we still act as if these are true.

You assume it is true, and write a program that relies on it.

Yeah, but every prime is the product of numbers!

All is number!

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