Defining the prime numbers prior to the integers: A first-principles approach to the distribution of primes

TL;DR

This paper introduces a novel first-principles approach to understanding prime numbers by defining them prior to natural numbers, offering new insights into their distribution, properties, and foundational significance.

Contribution

It proposes defining primes independently of natural numbers, providing new proofs, models, and an equivalent formulation of the Riemann hypothesis from this perspective.

Findings

New proof of the fundamental theorem of arithmetic

Probabilistic model explaining prime correlations

Equivalent formulation of the Riemann hypothesis

Abstract

While the prime numbers have been subject to mathematical inquiry since the ancient Greeks, the accumulated effort of understanding these numbers has - as Marcus du Sautoy recently phrased it - 'not revealed the origins of what makes the primes tick.' Here, we suggest that a resolution to this long-standing conundrum is attainable by defining the primes prior to the natural numbers - as opposed to the standard number theoretical definition of primes where these numbers derive from the natural numbers. The result is a first-principles perspective on the primes that exposes and explains the 'origins' of their distribution and their mathematical properties and provides an intuitive as well as pedagogical approach to the primes with the potential to impact our thinking about these age-old mathematical objects. A few immediate outcomes of this perspective are another proof of the fundamental theorem of arithmetic, a probabilistic model of primes sharing as well as explaining their subrandom correlation structure, and an equivalent formulation of the Riemann hypothesis.