Lei Mastrorocco
Year: 2025
Type: article
Abstract: The distribution of prime numbers has remained one of mathematics' most enduring mysteries since antiquity. From Euclid's proof of their infinitude to the modern Prime Number Theorem established by Hadamard and de la Vallée Poussin in 1896 [1], the mathematical community has sought to understand the underlying patterns governing these fundamental building blocks of arithmetic. While significant progress has been made in characterizing the asymptotic behavior of prime distribution through works such as Riemann's hypothesis [2] and various refinements of the Prime Number Theorem [3], the question of whether deterministic patterns exist in local prime distribution has remained largely unresolved. Traditional approaches to prime distribution analysis have generally assumed that primes exhibit pseudo-random behavior at local scales, with any apparent patterns being statistical artifacts rather than manifestations of underlying mathematical structure. This perspective has been reinforced by the apparent irregularity of prime gaps and the difficulty of predicting the location of individual primes beyond statistical approximations. However, recent advances in computational mathematics and the availability of extensive prime datasets have opened new possibilities for investigating potential patterns that may have been previously undetectable. The General Harmonic Law of Primes, which we designate as Mastrorocco's Law in recognition of its original formulation, proposes a fundamentally different perspective on prime distribution. Rather than viewing primes as exhibiting pseudo-random behavior, this law suggests that prime numbers follow an underlying organizational structure based on discrete symmetries and modular harmony that becomes increasingly apparent at larger scales. This theoretical framework introduces concepts of harmonic blocks, modular symmetry, and angular regularity that provide quantitative measures for detecting and characterizing harmonic patterns in prime sequences. The significance of this work extends beyond theoretical mathematics into practical applications in cryptography, where understanding prime distribution patterns could have profound implications for the security of current cryptographic systems. Modern public-key cryptography, particularly RSA encryption, relies fundamentally on the computational difficulty of factoring large composite numbers into their prime factors [4]. If deterministic patterns exist in prime distribution that can be exploited algorithmically, this could potentially impact the security assumptions underlying these systems. Our investigation represents the most comprehensive computational validation of a prime distribution theory ever undertaken, encompassing scales from hundreds of millions to trillions of numbers. Through innovative statistical sampling methodologies and distributed computational approaches, we have been able to test Mastrorocco's Law across scale factors that would have been computationally prohibitive using traditional exhaustive methods. The results demonstrate remarkable consistency and stability of harmonic metrics across these extreme scales, providing compelling evidence for the existence of fundamental harmonic structure in prime distribution. The implications of these findings extend beyond number theory into broader questions about the nature of mathematical structure and the relationship between apparent randomness and underlying order. The demonstration that harmonic patterns become more pronounced rather than less pronounced at larger scales challenges conventional expectations based on statistical fluctuation theory and suggests that prime numbers may exhibit emergent organizational properties that manifest most clearly at trillionic scales.
Author: MASTROROCCO, ANDRE
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6949660714
FWCI:
Cites:
Cited by:
Related to: