The Valence Layer Algorithm: A Structured Approach to Consecutive Prime Search and Its Implications for the Riemann Hypothesis
Year: 2026
Type: article
Abstract: This paper presents the Valence Layer Algorithm, a deterministic and efficient method for finding the next prime number from a given prime. The algorithm uses a fixed toolset of even numbers (valences) to decompose the gap between consecutive primes, ensuring rapid convergence and scalability to arbitrarily large numbers. Key contributions: A greedy decomposition algorithm for prime gaps using a toolbox of even numbers: {120, 64, 34, 20, 10, 6, 4, 2}. Proof of completeness: since 2 is in the toolbox, every even gap can be decomposed. The Cardinality Rule: the number of tools used is always ≤ G/2. Numerical validation for primes near 10¹², showing all gaps are exactly decomposable. A connection between the even/odd symmetry of prime gaps and the Riemann Hypothesis via maximum entropy (p = 0.5). A complete Fortran implementation (Miller–Rabin deterministic primality test, 64/128-bit arithmetic, command-line interface). The algorithm is deterministic, scalable, and provides a conceptual bridge between elementary arithmetic and analytic number theory.
Author: Gualberto
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7168190027
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