7. Conjectures and theorems

2020 ◽  
pp. 112-129
Author(s):  
Robin Wilson
Keyword(s):  
Prime Numbers ◽  
Unique Factorization ◽  
The 1950S ◽  
Positive Integers ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

‘Conjectures and theorems’ investigates a number of topics, such as the distribution of prime numbers, and two unsolved problems, Goldbach’s conjecture and the twin prime conjecture. The factorization of positive integers into primes is unique, but this does not hold for certain other systems of numbers. A more in-depth look at unique factorization gives deeper results, including a proposed result of Gauss. Mathematicians in the 1950s and 1960s confirmed that he was correct, as shown in the so-called ‘Baker-Heegner-Stark theorem’.

scholarly journals A Study of Authenticated Communication Based on Magic Square and Goldbach’s Conjecture

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Group Learning ◽  
Prime Numbers ◽  
Learning Problems ◽  
Student Group ◽  
Magic Square ◽  
Innovative Method ◽  
Encryption Decryption ◽  
Novel Applications ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

Although the magic square is a historical and universal study, its progress has been limited, to numeric games, which is closer to digital games or word games, and lacks the connection with mainstream mathematics. Recently, its study has extended from exciting mathematical games to various novel applications, such as image encryption, decryption processing, watermarking solutions, and student group learning problems, or different engineering applications. In terms of employment in information security, it is the blue ocean that requires more innovative research to enrich its content. In this study, we engage the magic square and Goldbach’s Conjecture to develop an innovative method to search prime numbers

scholarly journals On the Universal Encoding Optimality of Primes

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3155
Author(s):  
Ioannis N. M. Papadakis
Keyword(s):  
Linear Programming ◽  
Integer Linear Programming ◽  
Optimization Problem ◽  
Prime Numbers ◽  
Prime Factors ◽  
Universal Quantum ◽  
Optimality Properties ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

The factorial-additive optimality of primes, i.e., that the sum of prime factors is always minimum, implies that prime numbers are a solution to an integer linear programming (ILP) encoding optimization problem. The summative optimality of primes follows from Goldbach’s conjecture, and is viewed as an upper efficiency limit for encoding any integer with the fewest possible additions. A consequence of the above is that primes optimally encode—multiplicatively and additively—all integers. Thus, the set P of primes is the unique, irreducible subset of ℤ—in cardinality and values—that optimally encodes all numbers in ℤ, in a factorial and summative sense. Based on these dual irreducibility/optimality properties of P, we conclude that primes are characterized by a universal “quantum type” encoding optimality that also extends to non-integers.

scholarly journals Exploring Goldbach’s conjecture, the Pebonacci sequence and its five-dimensional vortex structure based on the logarithm of the circle

2020 ◽  
Vol 7 (8) ◽  
pp. 398-408
Author(s):  
Yiping Wang
Keyword(s):  
Mathematical Model ◽  
Closed Interval ◽  
Three Dimensional ◽  
Prime Numbers ◽  
Quadratic Equation ◽  
Space Structure ◽  
Number Sequence ◽  
Number Series ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

A method based on circle logarithm to prove Goldbach’s conjecture and Pebonacci sequence is proposed. Its essence is to deal with the real infinite series, each of the finite three elements (prime numbers, number series) has asymmetry problems, forming a basic even function one-variable quadratic equation and odd function one-variable three-dimensional number sequence; it is converted to "The irrelevant mathematical model expands latently in a closed interval of 0 to 1," forming a five-dimensional vortex space structure.

scholarly journals Proof of the Goldbach's Conjecture

2021 ◽  
Author(s):  
K.H.K. Geerasee Wijesuriya
Keyword(s):  
Prime Numbers ◽  
Goldbach’S Conjecture ◽  
Mathematics Problem ◽  
Goldbach's Conjecture

Goldbach’s Conjecture states that every even number greater than 3, can be written as a summation of two prime numbers. This conjecture is roughly 300 years old and a very famous unsolved mathematics problem. To prove the Goldbach’s Conjecture, I use the contradiction method in mathematics as below.

scholarly journals Vertex connectivity of the power graph of a finite cyclic group II

2019 ◽  
Vol 19 (02) ◽  
pp. 2050040 ◽  
Author(s):  
Sriparna Chattopadhyay ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Cyclic Group ◽  
Undirected Graph ◽  
Prime Numbers ◽  
Vertex Connectivity ◽  
Induced Subgraph ◽  
Power Graph ◽  
Finite Cyclic Group ◽  
Minimum Number ◽  
Positive Integers ◽  
Appl Math

The power graph [Formula: see text] of a given finite group [Formula: see text] is the simple undirected graph whose vertices are the elements of [Formula: see text], in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity [Formula: see text] of [Formula: see text] is the minimum number of vertices which need to be removed from [Formula: see text] so that the induced subgraph of [Formula: see text] on the remaining vertices is disconnected or has only one vertex. For a positive integer [Formula: see text], let [Formula: see text] be the cyclic group of order [Formula: see text]. Suppose that the prime power decomposition of [Formula: see text] is given by [Formula: see text], where [Formula: see text], [Formula: see text] are positive integers and [Formula: see text] are prime numbers with [Formula: see text]. The vertex connectivity [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl. 17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for [Formula: see text], we give a new upper bound for [Formula: see text] and determine [Formula: see text] when [Formula: see text]. We also determine [Formula: see text] when [Formula: see text] is a product of distinct prime numbers.

scholarly journals Asymptotic counting theorems for primitive juggling patterns

2019 ◽  
Vol 15 (05) ◽  
pp. 1037-1050
Author(s):  
Erik R. Tou
Keyword(s):  
Prime Number ◽  
Generating Functions ◽  
Arbitrary Number ◽  
Mathematical Theory ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Theoretical Framework ◽  
Positive Integers ◽  
Infinite Set ◽  
Fixed Period

The mathematics of juggling emerged after the development of siteswap notation in the 1980s. Consequently, much work was done to establish a mathematical theory that describes and enumerates the patterns that a juggler can (or would want to) execute. More recently, mathematicians have provided a broader picture of juggling sequences as an infinite set possessing properties similar to the set of positive integers. This theoretical framework moves beyond the physical possibilities of juggling and instead seeks more general mathematical results, such as an enumeration of juggling patterns with a fixed period and arbitrary number of balls. One problem unresolved until now is the enumeration of primitive juggling sequences, those fundamental juggling patterns that are analogous to the set of prime numbers. By applying analytic techniques to previously-known generating functions, we give asymptotic counting theorems for primitive juggling sequences, much as the prime number theorem gives asymptotic counts for the prime positive integers.

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