God made the integers…-Kronecker

As an immediate corollary of Chaitin’s theorem that for almost all integers $$n \in \mathbb{N}$$:

$$K_U(n) \sim \log_2(n)$$

we may use the unique prime factorisation theorem to deduce that all the information in the integers is stored in the prime numbers. That is, for almost all $$n \in \mathbb{N}$$:

$$n = \prod_{i=1}^k p_i^{\alpha_i} \implies K_U(n) \sim \sum_{i=1}^k \alpha_i \cdot \log_2 p_i$$

which is consistent with the Law of Conservation of Information. Thus, Chaitin’s theorem that almost all integers are incompressible provides us with a non-trivial information-theoretic interpretation of the unique prime factorisation theorem.