Overview

Prime numbers have fascinated mathematicians for centuries because of their apparent irregularity. But where does this irregularity really come from? This book proposes a radical change of perspective. Instead of searching for new formulas or algorithms to generate prime numbers, it focuses on the structure of composite numbers. Through polynomial families defined on discrete domains, the set of odd composite numbers is described in a complete, constructive, and geometrically interpretable way. Prime numbers emerge not as objects generated by a rule, but as structural residues: the values left out by the ordered structure of composites. The analysis clarifies why quadratic polynomial sieves represent a natural and insurmountable efficiency limit, why higher-degree constructions are structurally redundant, and why sub-linear approaches cannot be complete. The apparent chaos in the distribution of primes is not attributed to randomness or analytic pathologies, but to the discrete nature of arithmetic and to the complementarity between primes and composites. In the final chapter, the Riemann zeta function is interpreted as a global descriptive tool for this structural chaos: not a generator of prime numbers, but a lens through which their collective fluctuations become observable. A rigorous and reflective work for readers interested not in promises, but in understanding the true structural limits of arithmetic.

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