A Power Law Governing Prime Gaps

Matsushita, Raul; Silva, Sergio Da

doi:10.4236/oalib.1102989

Cited by 3 publications

(4 citation statements)

References 9 publications

(7 reference statements)

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“…Observe that most P(k) decay as a power law in these plots. Power laws have been found in the distribution of prime gaps [22] and in a myriad of contexts, such as a psychiatric ward [23] and financial crashes [24]. Here, P(k) is proportional to k taken to the power −c, that is, P(k) � Ak − c , in which A is the proportionality constant.…”

Section: Methodology and Numerical Resultsmentioning

confidence: 99%

A Numerical Study on the Regularity of d-Primes via Informational Entropy and Visibility Algorithms

Mayer

Monteiro

2020

Complexity

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Let a d-prime be a positive integer number with d divisors. From this definition, the usual prime numbers correspond to the particular case d=2. Here, the seemingly random sequence of gaps between consecutive d-primes is numerically investigated. First, the variability of the gap sequences for d∈2,3,…,11 is evaluated by calculating the informational entropy. Then, these sequences are mapped into graphs by employing two visibility algorithms. Computer simulations reveal that the degree distribution of most of these graphs follows a power law. Conjectures on how some topological features of these graphs depend on d are proposed.

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Section: Methodology and Numerical Resultsmentioning

confidence: 99%

A Numerical Study on the Regularity of d-Primes via Informational Entropy and Visibility Algorithms

Mayer

Monteiro

2020

Complexity

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“…Matsushita and Da Silva's study [1] analyzed prime gaps (intervals between consecutive primes) and found they follow a power law, suggesting a non-random pattern. This pattern indicates prime gaps are inversely related to a number's likelihood of being prime.…”

Section: Proposed Workmentioning

confidence: 99%

“…This connection is key to understanding prime distribution, as shown in the Prime Number Theorem, which connects the count of primes less than a number x to x/log(x). Linking the non-random prime gaps [1] to the Riemann Hypothesis, which focuses on the zeros of the Zeta function and prime distribution, is complex. While prime gap analysis is insightful, directly connecting it to the Hypothesis needs a new theoretical framework or math approach that relates these gaps to the Zeta function's zeros.…”

Section: Proposed Workmentioning

confidence: 99%

The Riemann Hypothesis: A Fresh and Experimental Exploration

Silva

2024

JAMCS

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This research proposes a new approach to the Riemann Hypothesis, focusing on the interplay between prime gaps and the non-trivial zeros of the Riemann Zeta function. Utilizing various statistical models and experimental analysis techniques, three important insights are uncovered: 1) Granger causality tests reveal a predictive relationship in which past non-trivial zeros may predict future prime gaps; 2) Complex, nonlinear interactions between prime gaps and non-trivial zeros are identified, challenging simple linear correlations; and 3) Causal network analysis reveals intricate feedback-loop relationships. These findings contribute to a better understanding of prime number distribution and the Zeta function, opening up novel possibilities for further mathematical research. The study aims to motivate mathematicians towards a proof or disproof of the Riemann Hypothesis.

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“…In fact in dealing with prime numbers one cannot miss to study their finite or discrete differences or gaps what has been done by some authors [35][36][37][38] though just at the first order i.e. Pn+1-Pn both in a classical i.e.…”

Section: Finite Numerical Data Setsmentioning

confidence: 99%

Statistical Distributions of Prime Number Gaps

Lattanzi

2024

JAMCS

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A heuristic i.e. empirical approach to the problem of prime number gaps of many kinds and types, different degrees and orders, treated as simple raw experimental data from the statistical viewpoint is presented. The aim of the article is to show a picture of the actual situation of prime number gaps in order to describe and to try to understand the structure itself of prime gaps of various kinds and orders as well as of primes themselves. The data base comprises the finite sequences of prime number gaps up to the value Pn of the prime counter n = 5∙107 that is P5E7 = P(5∙107) = 982,451,653 all of them available in the net. The statistical distributions of prime gaps are best-fitted by the pseudo-Voigt fit function, a convolution of the Lorentz and the Gauss differential distribution functions, or by the so-called E-exp or exp-exp differential distribution function or by a log-linear histogram according to the kind of gaps examined, either δiPn (higher order gaps) or ΔkPm = Pm – Pm–k (delta-lags) with i and k ≥ 2 or the simple linear differences δ1Pm = Δ1Pm = ΔPm= Pm – Pm–1 respectively. One of the unexpected results of the investigation is the appearance of inner structures at high values of nΔ, the number of the intervals of the distributions, suggesting the presence of groups or clusters strictly linked to the nature of prime numbers themselves in which the same phenomenology is present.

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A Power Law Governing Prime Gaps

Cited by 3 publications

References 9 publications

A Numerical Study on the Regularity of d-Primes via Informational Entropy and Visibility Algorithms

A Numerical Study on the Regularity of d-Primes via Informational Entropy and Visibility Algorithms

The Riemann Hypothesis: A Fresh and Experimental Exploration

Statistical Distributions of Prime Number Gaps

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