MAX Prime Theory (Mathematical Approach to eXploring Primes)

In this work, we propose the MAX Prime Theory—a novel and rigorous method for generating prime numbers that significantly increases the candidate density compared to classical approaches. The method is built on several key components:

Sequence A:
Candidates are generated using the function:
x = 25 + 5 · n(n + 1)
which produces the initial set of candidate numbers.

Transformation:
Each candidate x is transformed via the function:
f(x) = (6x + 5) / x
to produce a pair (N, d) with N = 6x + 5 and d = x. This ensures that N is of the form 6k + 1, a necessary condition for primality.

Modular Filters:
Preliminary congruence conditions (e.g., n ≡ 0 (mod 3) and n ≡ 3 (mod 7)) are applied to eliminate unproductive candidates. These filters are systematically extended to include primes up to 37 and are combined via the Chinese Remainder Theorem to select a subset of "fertile" candidates.

Enrichment Factor:
In this optimized subset, the probability of identifying prime numbers is amplified relative to the classical estimate of 1/ln(x). Experimental results show an enrichment factor F typically between 4.5 and 12, with some cases exceeding 18. A detailed asymptotic analysis confirms that as more modular filters are applied, F increases progressively.

Highlights:

• Rigorous theoretical analysis and formal proofs demonstrate that the iterative application of modular filters reduces the number of unproductive candidates and significantly improves computational efficiency.
• The method is invariant under small variations of the initial parameters.
• Unlike traditional sieve methods that eliminate multiples a posteriori, MAX Prime Theory employs a preventive screening strategy to optimize candidate selection.

Applications:
This approach offers new insights for developing more efficient prime number generation techniques, with potential applications in cryptography and in the search for large primes. All results are fully replicable, and further details are available at max-russo.com.