Entropy Primes and Integer Composites: The 2nd Quantization Approach

Abstract

Primes constitute a subset of all integers. Integers that are not prime are composite numbers, constituted of primes. Studies of numbers have been ongoing for millenia, and with recent advances making large strides forward in understanding and in connections between various viewpoints and approaches, yet we are unsure (as consensus) of whether there are patterns to primes or if they are uncorrelated random occurrences on the space of integers positive and infinite. However advances of the research yet if not the results sought for yet do occur, with the very question of determinism of occurrence of primes and its antipode of randomness of occurrence of primes prompting introduction of probability, and with probability the concept of entropy or perhaps the sequence is reversed for some researchers. In this letter we pursue such an investigation from entropy and randomness or statistics considerations. We approach this from discrete or quantum statistics which we argue are naturally number theoretic mappings or representation efficient and compact formalisms. We show how occupation number formalism or second quantization naturally reproduces recent entropy of numbers formulations.