Number Theory Problem Is Solved: Mathematicians are elated by a surprising proof of a famous and important conjecture

Science ◽  
1983 ◽  
Vol 221 (4608) ◽  
pp. 349-350 ◽  
Author(s):  
G. KOLATA
Keyword(s):  
Number Theory ◽  
Theory Problem

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 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.

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

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).

A Number Theory Problem from Multiple-Access Adder Channels

SIAM Review ◽  
1995 ◽  
Vol 37 (2) ◽  
pp. 240-241
Author(s):  
D. B. Jevtić
Keyword(s):  
Number Theory ◽  
Multiple Access ◽  
Theory Problem

Multiplicative Number Theory I

2006 ◽  
Author(s):  
Hugh L. Montgomery ◽  
Robert C. Vaughan
Keyword(s):  
Number Theory

A Pathway Into Number Theory

1996 ◽  
Author(s):  
R. P. Burn
Keyword(s):  
Number Theory
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