• Group primes by their grading values rather than congruence classes
• Analyze the distribution of gradings among primes
• Connect multiplicative properties of L-functions to additive structures
9.2 Prime Constellations and Twin Primes
The DULA framework naturally extends to studying prime constellations. For twin primes
(p, p + 2), both must avoid certain congruence classes. The grading functions can help
characterize admissible patterns.
Proposition 9.1 (Twin Prime Constraint). For twin primes (p, p + 2) with p > 3, if p ≡ 5
(mod 6), then p + 2 ≡ 1 (mod 6). In terms of DULA gradings modulo 6:
• ϕ
6
(p) = 1 (contributing to the odd class)
• ϕ
6
(p + 2) = 0 (contributing to the even class)
This asymmetry in gradings reflects the inherent structure of twin prime pairs.
9.3 Computational Applications
• Character evaluation: For small moduli where discrete logs are feasible, DULA
gradings provide efficient character evaluation
• Factorization patterns: The grading functions can detect certain factorization pat-
terns without full prime decomposition
• Pseudoprimality testing: Grading-based tests could supplement existing primality
criteria
9.4 Open Problems and Extensions
1. Quantitative density results: Can DULA gradings provide new bounds on the
density of primes in arithmetic progressions?
2. Higher-dimensional analogues: Extend the framework to number fields and their
units groups
3. Effective bounds: For which moduli k can the discrete logarithms in the grading
functions be computed efficiently?
4. Connection to class field theory: How do DULA gradings relate to Artin reci-
procity and class field towers?
5. Analytic number theory: Can the additive structure revealed by DULA gradings
lead to new estimates for error terms in the prime number theorem for arithmetic
progressions?
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