A New and Deterministic Scheme for Characterizing The Organization of Prime Numbers

Abstract : The fundamental theorem of arithmetic states that any composite natural integer can be expressed in one and only one way as a product of prime numbers. This sets the understanding of the organization of prime numbers at the core of number theory. In this work we present a simple, self-consistent and deterministic scheme allowing to investigate further the intrinsic organization of prime numbers. Using this scheme, we establish an algorithm that yields the complete list of prime numbers below any preassigned limit x. Counting the latter yields π(x), the number of prime numbers below x. Based on preliminary tests on computing clusters available, a considerable gain in computational speed and algorithmic simplicity towards producing complete lists of large prime numbers is observed. At the core of the new scheme lays its ability to provide, in a deterministic way, complete lists of consecutive and composite odd numbers below any preassigned limit x. The complete list of prime numbers below x is deduced from the latter. The two key ingredients of the scheme are a set of eleven generic tables, coupled with a three-criteria test applied on the differences between pairs of the consecutive composite odd numbers initially obtained. Since it leads to counting all the elements of a complete list of prime numbers up to x, our deterministic scheme provides a new approach to the long standing problem of " how many prime numbers are there below any preassigned limit x ". The said scheme therefore potentially contributes towards studies aimed at unveiling the organization of prime numbers. We illustrate the latter in a follow-up paper, Paper II [3], where we propose a new perspective on the Riemann hypothesis.
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Guilaume Hawing, Patrice Okouma. A New and Deterministic Scheme for Characterizing The Organization of Prime Numbers. 2016. ⟨hal-01407334⟩

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