Prime numbers are often defined as the most mysterious figures in mathematics and they have been being studied for almost 2500 years, yet we haven't fully understood what their nature and structure consists on. So I think I'm right by stating that there's no such thing as a formula that exhaustively and precisely describes the seemingly random distribution of primes.

So my questions are:

  1. What would be the impact of the discovery of such formula? Would the its finding be of great importance? I assume so, since it would pretty much solve the prime number mystery, right? Could you please compare it with another discovery?
  2. Would the discovery of an easily computable formula that generates every single prime number be relevant?

It turns out that opinions (there shouldn't be, there should be a definite answer since they're not subjective questions) differ widely depending on the site you're on, and a quality reply would be greatly appreciated.

Thank you in advance, Dan

  • $\begingroup$ Just a comment: the distribution of primes is not completely random. The Prime Number Theorem tells us something about the asymptotic distribution of primes, namely that the number of primes less than $x$ approaches $li(x)$, the logarithmic integral. Granted, we still don't know where primes are exactly, but there is some global structure there. $\endgroup$ – bartgol Mar 31 '16 at 17:35
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    $\begingroup$ $1.$ is subjective in that people may differ in how they describe it's impact, and also their choice of comparison to other discoveries. $\endgroup$ – snulty Mar 31 '16 at 17:35
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    $\begingroup$ I don't agree that question 1. is not subjective. What does "exhaustively and precisely" mean? If that means that that formula would be enough for being able to solve any formal question about prime numbers, then probably that formula would end with what is known by "number theory" now. $\endgroup$ – Z. L. Mar 31 '16 at 17:36

Concerning 1., the proof of the optimal error term for the distribution of prime numbers, namely $$ \pi(x) = \text{Li}(x) + O(\sqrt x \log x) $$ would be a great discovery for whole number theory, since it would prove (in fact, be equivalent to) the Riemann hypothesis.

Concerning 2., it has been discussed many times that such a formula does not exist - see here. Of course, there is a discussion what we want to consider as such a formula, because there are also useless ones, see here.


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