# Projects

# 1. The Monte Carlo Hypothesis

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.

# 2. Koopman Open

The primary objective of this challenge in experimental mathematics is to query an information-theoretic lower-bound on the computational complexity of integer factorisation. Due to a natural correspondence between integer factorisation and unbounded Koopman operators, this problem is reducible to neural program synthesis for approximating the eigenfunctions of a suitable Koopman operator.