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There has been a lot of research and failed attempts to establish the pattern in prime numbers. However, can it be established once and for all that there may be no set pattern that might be exhibited by the distribution of prime numbers? Maybe the problem is in essence like finding solvable extensions in the form of radicals for the quintic polynomials wherein no such solvable extension exist.

So instead of establishing for a pattern for the distribution of prime numbers, can it be axiomised or better still, proved that prime number distribution is just random even though the Gauss distribution formula or the Riemann-Zeta function may come close to establishing distribution pattern but ultimately taking a larger set of prime number, no distribution formula or pattern exists?

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closed as too broad by Saad, zhoraster, TheSimpliFire, Claude Leibovici, Did Mar 15 '18 at 8:29

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    $\begingroup$ Certainly it is known that primes are more sparsely distributed among larger integers than among smaller ones (The Prime Number Theorem), so there cannot be any "periodic" pattern of prime numbers. However the Riemann Zeta function provides an exact correction to the approximate distribution/prime counting function given by Gauss's logarithmic integral. So in that sense we have to be more careful in describing what a "pattern in prime numbers" consists of. $\endgroup$ – hardmath Mar 15 '18 at 4:15
  • $\begingroup$ Terence Tao has some interesting notes on the randomness of the distribution of primes: terrytao.files.wordpress.com/2009/07/primes1.pdf $\endgroup$ – Tob Ernack Mar 15 '18 at 4:57
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    $\begingroup$ The word "pattern" requires a strict definition. Otherwise the question cannot be answered. Astonishing enough that there is a polynomial in $26$ variables for which the positive integer values are exactly the primes. $\endgroup$ – Peter Mar 15 '18 at 8:43
  • $\begingroup$ @Peter: at the risk of being corrected, because I haven't checked--isn't it a subset of the primes? $\endgroup$ – daniel Mar 15 '18 at 14:37
  • $\begingroup$ @daniel No, EVERY prime is a possible value. The values not just form a (strict) subset. $\endgroup$ – Peter Mar 15 '18 at 14:42