By what main criteria does the mathematical community judge whether an open problem is important among many others open problems in mathematics? For example if we say that the Riemann Hypothesis is an important problem because it is related to the prime distributions, are all open problems related to prime distributions important?

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    Terry Tao wrote an article What is good mathematics? that might be insightful for you. – Kimball Dec 26 '14 at 0:59
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    @ KimballThat is an excellent article, indeed, though half of the mathematical community I know was rolling on the floor laughing because of the fact that the main example of "good mathematics" there was the one Terry worked on for the last few years and the other half was "somewhat perplexed" by it. I just discard the fact because it is natural to draw the examples from what is currently on your mind and because everyone can get carried away a bit when writing (that is certainly true as far as I am concerned). – fedja Dec 26 '14 at 13:03
  • @fedja, for example why RH is important then Goldback conjecture however the 2 topic are about primes ? – zeraoulia rafik Dec 28 '14 at 18:42
  • @user51189 1) "Both being about primes" is not any more meaningful comparison than "both being about numbers" 2) It is not RH, which is important, but the validity of RH (a counterexample will result just in a spectacular collapse of a huge body of conditional statements) and 3) Have I ever said in public or in private that Goldbach conjecture is less important than RH? – fedja Dec 28 '14 at 19:09
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    @user51189 My #3 was just a way to attract your attention to the fact that if I had asked you "Why are flowers more important than crocodiles?", you would be in the same position as I was when looking at your question as posed. How am I supposed to answer a why question about the statement that I have trouble even assigning a clear meaning to, not mentioning agreeing with? :-) – fedja Dec 28 '14 at 21:46

An open problem is important if it's resolution affects many, diverse areas (indeed, the farther flung the impact, the more significant the problem).

In the case of the Reimann hypothesis, there are many theorems that are conditional on it (see Consequences of Riemann hypothesis), though they are largely in the broader realm of number theory.

Other problems, like the P vs NP problem could have consequences beyond automata theory that include cryptography (and, by extension, our entire e-commerce industry), computational biology, manufacturing and optimization, etc.

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