Logarithmic Self-Symmetry of Prime Numbers — Theoretical Foundations and Computational Evidence

This work presents a theoretical and computational study on the logarithmic self-symmetry of prime numbers. The research introduces a geometric interpretation of multiplication, showing that every integer can be represented in a mirrored logarithmic space around its square root. This approach reveals a pattern of multiplicative symmetry that applies to both composite and prime numbers. Using large-scale computations of consecutive prime ratios, the study demonstrates that the logarithmic increments between consecutive primes tend to stabilize as numbers grow larger. This statistical stabilization suggests a possible invariant distribution of prime ratios, consistent with the Prime Number Theorem. The work proposes a new geometric framework for visualizing prime distribution and a conjecture on logarithmic invariance of prime gaps. It unifies multiplicative structure and logarithmic growth under a single theoretical model. Author: Luiz Alessandro Bittencourt MochkoAffiliations: Universidade Federal do Paraná (UFPR); UniEnsino, BrazilRole: ProducerLicense: Creative Commons Attribution 4.0 (CC BY 4.0)Date: October 2025