If we assume that prime number distribution is COMPLETELY random (subject to 1/log(x) restriction), can we prove twin prime conjecture or Goldbach conjecture ?

My feeling is that, this will be trivial for twin prime conjecture, but how to give a rigorous proof ? Does "completely random" imply that there will be infinite twin primes ?

On the other hand, if prime number distribution is completely random, will Goldbach conjecture still hold true ?

For example, if we consider all numbers between 1 and 100, if we assume prime number completely random distribution, will Goldbach conjecture still hold true For this set of numbers ?

The reason I ask this question, because, people often said that "primes behave almost completely random except subject to 1/log(x) restriction".


closed as off-topic by Peter Humphries, Marco Golla, Boris Bukh, GH from MO, Will Jagy Nov 9 '15 at 20:08

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  • $\begingroup$ 'Almost completely random' except that, for example, the probabilities of $p$ and $p+2$ being simultaneously prime are not independent. $\endgroup$ – Sylvain JULIEN Nov 9 '15 at 17:20
  • 2
    $\begingroup$ If we assumed prime distribution is COMPLETELY random, then we would have infinitely many pairs of primes $p,p+1$. Indeed, we would have arbitrarily many long consecutive runs of primes. $\endgroup$ – Wojowu Nov 9 '15 at 18:02

This is well documented in many places. See for example: https://terrytao.wordpress.com/2015/01/04/254a-supplement-4-probabilistic-models-and-heuristics-for-the-primes-optional/


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