Last modified at: 26-04-2021
The challenge is to explore the hypothesis that prime-encodings are algorithmically random and that prime numbers have a maximum entropy distribution. Conventional mathematical wisdom, at present, suggests that this hypothesis might be false. There might be 'conspiracies' among arbitrarily large subsequences of prime encodings, i.e. predictable behaviour, which may be exploited by a machine learning algorithm.
In fact, this hypothesis implies a non-trivial interpretation of the prime number theorem. The Prime Number Theorem says how the prime numbers are distributed but not why. On the other hand, an information-theoretic analysis of the Prime Number Theorem indicates that they are distributed in this way because prime encodings are algorithmically random and the prime numbers have a maximum entropy distribution.
It is not possible to prove that a particular object is incompressible within algorithmic information theory so the best we can do is perform rigorous experimental analysis using machine learning methods. Hence this challenge.