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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".

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

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Marco Golla, GH from MO, Will Jagy
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ '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
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    $\begingroup$ 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
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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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