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I am trying to write a piece of future fiction where one of the characters is famous for proving an important truth related to primes. I want to make it as realistic as possible, but i'm not a math major, and I need some guidance.

Basically the "function" takes as any input an real number and returns (in a single step) a real number related to prime numbers. so an integer like "3" returns the prime number "5", and something like "3.5" returns "6", a real number between prime #3 (5) and prime #4 (7).

The function is also reversible, so any number can (again in a single step) be determined to be prime. e.g. input of "29" returns "10", and because that's an integer we know it's prime, and the 10th prime. It would work both ways for any size of input number.

What would be the downstream effects of such a "function" being proven true? How would this kind of "function" open up new avenues of research in math? Would something like this be more obscure and only known to mathematicians, or known broadly to the general public, or somewhere in between, like Poicare's theorem?

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It would be the most amazing theorem of all time. It may open up new avenues trying to generalize the result, but it would also certainly close down an enormous area of research. –  Cass Sep 6 '13 at 3:47
    
Something probably worth looking into for your fiction is that (and don't press me about this, because I don't really know anything about it) primality is somehow related to data encryption. It may be that such a function would make encrypted data very easy to decrypt, which would obviously have huge global consequences, but like I say, I don't really know. –  Cass Sep 6 '13 at 3:53
    
Oh wait, sorry, I skimmed your post and mainly noticed the part where you said "any number can be determined to be prime in a single step." I don't think your function, as described in the previous paragraph, does that. My commentary was related to a 1-step plug-in machine which tells you immediately whether a number is prime or not. –  Cass Sep 6 '13 at 3:56
    
Even now we can calculate such functions, probably, by very long complex calculation for big numbers. So I think that the devising of such a function can be firstly useful in applications, but it will be a great discovery in pure math provided this function will have a specific or simple form, for instance, allowing to solve come classical problems, like Goldbach conjecture or infiniteness of the number of prime twins. –  Alex Ravsky Sep 6 '13 at 11:53
    
@Cass - I'm not a mathmetician, so I'm sure my description has all kinds of holes in it. I was hoping to use it as a plot point involving encryption. I don't want to make a really obvious and silly mistake –  SteveED Sep 7 '13 at 0:10
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Since it is science fiction we have a certain free space for speculation.

It must be clear, that no function using common elements can be sufficient. So possibly one comes up with a vector of, say, 9 additional components which may freely added to the definition of complex numbes, such that we have not only the two components real and complex of a numbers, but also some index denoting a sophisticated selected branchcut of the logarithm of this number, and 8 more components counting up to an overall of 11 components of a complex number where 9 are depending on the choice of the underlying universe.

The guy who has found this had made some educated guesses for parameters when simulating the evolution of various types of universes on bases of the 11-dimensional stringtheory. Here he observed at one time a pattern in the concentration of the universum-specific Riemann-gases and this gave then the values for the missing 8 parameters (besides the selection of the logarithmic branchcut).

After that, he could define a direct formula for the zeros of the Riemann-zeta-function which included the proof of the Riemann-hypothesis.

Having this powerful tool there were formulae possible, with which zeta-regularization, exact computation of quantum-effects and prognosis for the Brownian motion of any set of elementary particles over about 5 seconds was possible. Unfortunately that formula allowed computation only for elementary-particle sets of prime-number cardinality - improvements for generalizations are expected.

In cosmology all the main formulae could be revisited and reformulated in terms of the 9 additional components of the definition of a complex number, such that we see, that not only the cosmological process should be formulated in terms of functional iteration with convergence to fixpoints, such that in the retroverse view even the time-parameter cannot be computed to a "begin" but only approximate negative infinity of that selfcomposing/iterational process. The position and motion-parameters of stellar objects, the planets and even of asteroids of a certain mass could now be computed and prognosed for more than 100 of iterations into the future (where 10000 such iterations mark one "Kalpa" (or eternal periods) whose first knowledge was transmitted to us by the enlightened Buddha) and this initiated three international projects, which prepare a defense strategy against the three asteroid-collisions which have now to be expected in the next 175 years - projects which have a guaranteed financial base of 1% of the bsp of the 15 engaged nations and which thus generate also a complete new reliable employment perspective for all engaged scientists and other employees - stuff for a lot of sociologists to do research about and to initialize well funded studies.

Applied to biological processes like prime-ratio evolution schemes for pairs of species which are mutually destructive for the other one, retroversial computations of the species distribution in the continent in Australia for the recent 5 million years was possible and the prognosis into the next 1000 generation seems extreme reliable. For the prognosis of the greater landmasses bigger computer were needed but the planning in the ministries for ecological development has fundamentally been changed -in subject of research as well as in funding habits.

(...) ...

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Thank you very much! –  SteveED Sep 17 '13 at 1:20
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