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Related Concept Videos

Sequences01:29

Sequences

75
Sequences are fundamental mathematical objects consisting of ordered lists of numbers that follow a specific rule or pattern. Sequences are critical in various mathematical concepts, including calculus, series, and number theory. They can model real-world phenomena such as population growth, financial investments, and physical processes like the diminishing height of a bouncing ball.Each number in a sequence is referred to as a term. Typically, the terms are denoted as a1, a2, a3,…, where...
75
Mathematical Induction01:29

Mathematical Induction

106
Mathematical induction is a structured method of proof used to confirm the truth of statements involving natural numbers. Consider the sum of the first n natural numbers:This formula describes a pattern that appears to hold true as more terms are added. To verify that it is valid for all natural numbers, mathematical induction proceeds in two essential steps. The first is the base case, where the formula is tested for the initial value, typically n = 1. Substituting into both sides confirms the...
106
Arithmetic Sequences01:30

Arithmetic Sequences

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An arithmetic sequence is a structured arrangement of numbers where each term is derived by adding a constant value, known as the common difference, to the previous term. This consistent pattern allows for the efficient computation of any term within the sequence as well as the cumulative sum of multiple terms. The formula for finding the nth term of an arithmetic sequence is:Here, aₙ represents the nth term of the sequence, a is the first term, d is the common difference, and n is the...
72
Geometric Sequences01:30

Geometric Sequences

78
In systems where values diminish by a constant proportion at each stage, the resulting sequence follows a geometric structure. Each new value in the sequence is obtained by applying a fixed multiplier to the preceding term. This regular, proportional decline type is often used to represent processes involving gradual loss, such as energy dissipation or reduction in amplitude over time.When analyzing the total effect of such a process across unlimited iterations, the series of values is referred...
78
Determination of Pi Terms01:15

Determination of Pi Terms

456
The Buckingham Pi theorem is a valuable method in dimensional analysis, reducing complex relationships between variables into dimensionless terms. Relevant variables in analyzing the lift force on an airplane wing include lift force, air density, wing area, aircraft velocity, and air viscosity. Expressing each variable in terms of fundamental dimensions — mass, length, and time — provides a consistent foundation for constructing these dimensionless terms.
The theorem indicates that the...
456
Basic Discrete Time Signals01:16

Basic Discrete Time Signals

510
The unit step sequence is defined as 1 for zero and positive values of the integer n. This sequence can be graphically displayed using a set of eight sample points, showing a step function starting from n=0 and remaining constant thereafter.
The unit impulse or sample sequence is mathematically expressed as zero for all n values except at n=0, where it is one. The unit impulse sequence, denoted by δ(n), is the first difference of the unit step sequence, while the unit step sequence u(n) is the...
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On a Dynamical Approach to Some Prime Number Sequences.

Lucas Lacasa1, Bartolome Luque2, Ignacio Gómez2

  • 1School of Mathematical Sciences, Queen Mary University of London, Mile End, London E1 4NS, UK.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary

Dynamical systems theory reveals surprising patterns in prime number residues. Prime residue sequences exhibit maximal chaos and non-uniform block distributions, challenging number theory expectations.

Keywords:
chaoscomplex systemsentropyfractalsgap residuesnonlinearityprime numberssymbolic dynamics

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Area of Science:

  • Number Theory
  • Dynamical Systems Theory
  • Symbolic Dynamics

Background:

  • Number theory often studies prime numbers and their properties.
  • Dynamical systems theory provides tools to analyze complex sequences and patterns.

Purpose of the Study:

  • To explore patterns in prime number residue sequences using nonlinear and symbolic dynamics.
  • To investigate the chaotic nature and block distributions within these sequences.

Main Methods:

  • Analyzing prime number residues modulo k.
  • Examining prime gap residues.
  • Applying concepts from nonlinear and symbolic dynamics.
  • Calculating Renyi entropies and Kolmogorov-Sinai entropy.
  • Using the chaos game and Iterated Function Systems (IFS).

Main Results:

  • Prime residue sequences modulo k are maximally chaotic and lack forbidden patterns.
  • These sequences show non-uniform distributions of blocks (m>1), contrasting with Dirichlet's theorem.
  • Prime gap residue sequences are chaotic but exhibit weaker chaos with emerging forbidden patterns.
  • The distribution of admissible blocks in prime gap residues is non-uniform, supported by Hardy-Littlewood k-tuple conjecture.

Conclusions:

  • Cross-disciplinary transfer from dynamical systems to number theory yields fruitful insights.
  • Prime number sequences exhibit complex, non-uniform patterns not fully captured by traditional number theory.
  • Symbolic dynamics and chaos theory offer powerful frameworks for understanding prime number properties.