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

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