### A conceptual proposal on the undecidability of the distribution law of prime numbers and theoretical consequences

#### Abstract

*theoretical*unavailability and indemonstrability of the existence of a law of distribution of prime numbers. Tentatively, we conceptually consider demonstrability as computability, in our case the

*conceptual availability*of an algorithm able to compute the general properties of the presumed primes’ distribution law

*without computing such distribution*. The link between the conceptual availability of a distribution law of primes and decidability is given by considering how to

*decide*if a number is prime

*without*computing. The supposed distribution law should allow for any given prime knowing the next prime

*without factorial computing*.

*Factorial properties of numbers, such as their property of primality, require their factorisation (or equivalent, e.g., the sieves), i.e., effective computing. However, we have factorisation techniques available, but there are no (non-quantum) known algorithms which can effectively factor arbitrary large integers. Then factorisation is undecidable. We consider the theoretical unavailability of a distribution law for factorial properties, as being prime, equivalent to its non-computability, undecidability.*The availability and demonstrability of a hypothetical law of distribution of primes is inconsistent with its undecidability. The perspective is to transform this conjecture into a theorem.

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DOI: http://dx.doi.org/10.23755/rm.v37i0.480

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