scholarly journals The equivalent proposition of an unsolved number theory problem "is the difference between two primes"

2019 ◽  
Vol 9 (1) ◽  
pp. 1-2
Author(s):  
Mi Zhou ◽  
◽  
Shi Honwei ◽  
Zhang Delong ◽  
Jiang Xingyi ◽  
...  
Keyword(s):  
Number Theory ◽  
Theory Problem ◽  
The Difference

Density of sets with missing differences and applications

2021 ◽  
Vol 71 (3) ◽  
pp. 595-614
Author(s):  
Ram Krishna Pandey ◽  
Neha Rai
Keyword(s):  
Number Theory ◽  
Asymptotic Density ◽  
Theory Problem ◽  
Distance Graphs ◽  
Positive Integers ◽  
Infinite Set

Abstract For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.

scholarly journals Distance Metrics of D Numbers

2020 ◽  
Author(s):  
Liguo Fei ◽  
Yuqiang Feng
Keyword(s):  
Number Theory ◽  
Incomplete Information ◽  
Distance Function ◽  
Distance Measure ◽  
Complete Information ◽  
Belief Function ◽  
Distance Metrics ◽  
Modeling Methods ◽  
The Difference ◽  
D Numbers

Belief function has always played an indispensable role in modeling cognitive uncertainty. As an inherited version, the theory of D numbers has been proposed and developed in a more efficient and robust way. Within the framework of D number theory, two more generalized properties are extended: (1) the elements in the frame of discernment (FOD) of D numbers do not required to be mutually exclusive strictly; (2) the completeness constraint is released. The investigation shows that the distance function is very significant in measuring the difference between two D numbers, especially in information fusion and decision. Modeling methods of uncertainty that incorporate D numbers have become increasingly popular, however, very few approaches have tackled the challenges of distance metrics. In this study, the distance measure of two D numbers is presented in cases, including complete information, incomplete information, and non-exclusive elements

Index, sub-index and sub-factor of groups with interactions to number theory

2019 ◽  
Vol 19 (06) ◽  
pp. 2050101
Author(s):  
M. H. Hooshmand
Keyword(s):  
Number Theory ◽  
Direct Product ◽  
Prime Numbers ◽  
Additive Number Theory ◽  
Open Problems ◽  
Famous Conjecture ◽  
Future Directions ◽  
Left And Right ◽  
The Difference ◽  
Equivalent Conditions

This paper is the first step of a new topic about groups which has close relations and applications to number theory. Considering the factorization of a group into a direct product of two subsets, and since every subgroup is a left and right factor, we observed that the index conception can be generalized for a class of factors. But, thereafter, we found that every subset [Formula: see text] of a group [Formula: see text] has four related sub-indexes: right, left, upper and lower sub-indexes [Formula: see text], [Formula: see text] which agree with the conception index of subgroups, and all of them are equal if [Formula: see text] is a subgroup or normal sub-semigroup of [Formula: see text]. As a result of the topic, we introduce some equivalent conditions to a famous conjecture for prime numbers (“every even number is the difference of two primes”) that one of them is: the prime numbers set is index stable (i.e. all of its sub-indexes are equal) in integers and [Formula: see text]. Index stable groups (i.e. those whose subsets are all index stable) are a challenging subject of the topic with several results and ideas. Regarding the extension of the theory, we give some methods for evaluation of sub-indexes, by using the left and right differences of subsets. At last, we pose many open problems, questions, a proposal for additive number theory, and show some future directions of researches and projects for the theory.

scholarly journals Divisions by Two in Collatz Sequences: A Data Science Approach

2020 ◽  
Vol 21 ◽  
pp. 1-13
Author(s):  
Christian Koch ◽  
Eldar Sultanow ◽  
Sean Cox
Keyword(s):  
Number Theory ◽  
Data Science ◽  
Theory Problem ◽  
Mathematical Methods ◽  
Collatz Conjecture ◽  
Science Approach ◽  
Specific Length

The Collatz conjecture is an unsolved number theory problem. We approach the question by examining the divisions by two that are performed within Collatz sequences. Aside from classical mathematical methods, we use techniques of data science. Based on the analysis of 10,000 sequences we show that the number of divisions by two lies within clear boundaries. Building on the results, we develop and prove an equation to calculate the maximum possible number of divisions by two for any given a Collatz sequence. Whenever this maximum is reached, a sequence leads to the result one, as conjectured by Lothar Collatz. Furthermore, we show how many divisions by two are required for a cycle of a specific length. The findings are valuable for further investigations and could form the basis for a comprehensive proof of the conjecture.

EXACT AND SUPERLATIVE PRICE AND QUANTITY INDICATORS

2009 ◽  
Vol 13 (S2) ◽  
pp. 335-380 ◽  
Author(s):  
W. Erwin Diewert ◽  
Hideyuki Mizobuchi
Keyword(s):  
Number Theory ◽  
Unit Cost ◽  
Utility Functions ◽  
Cost Functions ◽  
Index Number ◽  
Economic Approach ◽  
Index Number Theory ◽  
Nonhomothetic Preferences ◽  
The Difference

The traditional economic approach to index number theory is based on a ratio of cost functions. Diewert defined superlative price and quantity indices as observable indices that were exact for a ratio of unit cost functions or for a ratio of linearly homogeneous utility functions. The present paper looks for counterparts to his results in the difference context, for both flexible homothetic and flexible nonhomothetic preferences. The Bennet indicators of price and quantity change turn out to be superlative for the nonhomothetic case. The underlying preferences are of the translation-homothetic form discussed by Balk, Chambers, Dickenson, Färe, and Grosskopf.

scholarly journals Alternative Branching Strategies in the Branch and Bound Algorithm by Using a K-Clique Covering Vertex Set for Maximum Clique Problems

2020 ◽  
Vol 1 (4) ◽  
pp. 208-216
Author(s):  
Mochamad Suyudi ◽  
Asep K. Supriatna ◽  
Sukono Sukono
Keyword(s):  
Lower Bound ◽  
Branch And Bound ◽  
Maximum Clique ◽  
Maximum Clique Problem ◽  
Theory Problem ◽  
Maximum Cardinality ◽  
Arbitrary Graph ◽  
Complete Subgraph ◽  
Vertex Set ◽  
The Difference

The maximum clique problem (MCP) is graph theory problem that demand complete subgraph with maximum cardinality (maximum clique) in arbitrary graph. Solving MCP usually use Branch and Bound (BnB) algorithm. In this paper, we will show how n + 1 color classes (where n is the difference between upper and lower bound) selected to form k-clique covering vertex set which later used for branching strategy can guarantee finding maximum clique.

A Number Theory Problem from Multiple-Access Adder Channels

SIAM Review ◽  
1993 ◽  
Vol 35 (3) ◽  
pp. 489-490
Author(s):  
D. B. Jevtić
Keyword(s):  
Number Theory ◽  
Multiple Access ◽  
Theory Problem

A Number Theory Problem Concerning Finite Groups and Rings

1964 ◽  
Vol 7 (1) ◽  
pp. 23-34 ◽  
Author(s):  
Ian G. Connell
Keyword(s):  
Number Theory ◽  
Finite Groups ◽  
Abelian Groups ◽  
Theory Problem ◽  
Structure Theorems

Let f1(n) denote the number of abelian groups of order n and f2(n) the number of semi-simple rings with n elements. What can be said about the magnitude of fi(n)? We shall prove that one can expect, on the average, about 2.3 groups and 2.5 rings of the kind stated for a given order. First we state without proof the two relevant structure theorems (which are readily available in standard texts).

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