📜 The Wessen Twin Prime Identity
Main Identity: Rmodular(pmaximum) = ¼ · Ctwin(pmaximum) · (Mno-two(pmaximum))3
Component Definitions:
• Rmodular(pmaximum) = ¼ ∏3 ≤ prime ≤ pmaximum [(prime-1)(prime-2)/prime²]
• Ctwin(pmaximum) = ∏3 ≤ prime ≤ pmaximum [1 - 1/(prime-1)²]
• Mno-two(pmaximum) = ∏3 ≤ prime ≤ pmaximum [1 - 1/prime]
Where the products are taken over all odd primes from 3 up to pmaximum inclusive.
🔍 Mathematical Context
This identity connects three fundamental aspects of number theory:
Rmodular: Represents modular residue ratios in prime-based sieves, encoding how twin prime candidates survive modular constraints.
Ctwin: Twin prime correlation factor derived from Hardy-Littlewood circle method, measuring pair correlations in prime distributions.
Mno-two: Modular "no-two" structure capturing exclusion principles in residue systems, related to Euler totient densities.
The identity bridges discrete modular arithmetic with analytic number theory, providing a finite-cutoff version of asymptotic twin prime conjectures.
⚡ Computational Verification Methods
Algorithm 1 (Numeric Mode): Uses high-precision floating-point arithmetic with configurable tolerance testing. Computes each component separately and verifies the identity through ratio analysis with |ρ - 1| ≤ ε criterion.
Algorithm 2 (Exact Mode): Employs exact integer arithmetic using BigInt operations to eliminate all floating-point precision errors. Tests the algebraically equivalent condition: N
R × D
C × D
M³ = D
R × N
C × N
M³ for absolute verification.
🎯 Research Implications
If Identity Holds Generally: Would provide the first exact finite formula for twin prime distributions, revolutionizing computational approaches to the Twin Prime Conjecture.
If Identity Fails Beyond pmaximum=5: Would indicate fundamental transition points in prime structure, potentially revealing where classical analytic methods break down and new approaches are needed.
Connection to Riemann Hypothesis: The modular structure relates to Möbius function cancellation discussed in the broader framework, providing computational testing grounds for RH-equivalent formulations.