A UNIQUE FACTORIZATION IN COMMUTATIVE MÖBIUS MONOIDS

2008 ◽  
Vol 04 (04) ◽  
pp. 549-561 ◽  
Author(s):  
EMIL DANIEL SCHWAB ◽  
PENTTI HAUKKANEN
Keyword(s):  
Fundamental Theorem ◽  
Factorization Theorem ◽  
Unique Factorization ◽  
Multiplicative Semigroup ◽  
Arithmetical Functions ◽  
Bicyclic Semigroup ◽  
Fundamental Theorem Of Arithmetic ◽  
Positive Integers ◽  
Special Case

We show that any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Particular attention is paid to standard examples, which arise from the bicyclic semigroup and the multiplicative analogue of the bicyclic semigroup. The second example shows that the Fundamental Theorem of Arithmetic is a special case of the unique factorization theorem in commutative Möbius monoids. As an application, we study generalized arithmetical functions defined on an arbitrary commutative Möbius monoid.

The board stretcher: A model to introduce factors, primes, composites, and multiplication by a fraction

1973 ◽  
Vol 20 (8) ◽  
pp. 649-656
Author(s):  
Louis S. Cohen
Keyword(s):  
Fundamental Theorem ◽  
Number System ◽  
Factorization Theorem ◽  
Inverse Relation ◽  
Unique Factorization ◽  
Composite Number ◽  
Prime Factors ◽  
Fundamental Theorem Of Arithmetic

Most children are aware that some numbers are even and the rest are odd, but they know little else about our number system. The “board-stretcher” model provides a means to discovering the idea of factors and obtaining a feeling for prime and composite numbers. It portrays the inverse relation between multiplication and division and, as a finale, it can be used to introduce the concept of multiplication by a fraction. This model suggests the notion that the sum of the prime factors of a number (except for four) is always less than the product of the factors. The Fundamental Theorem of Arithmetic (or Unique Factorization Theorem) is understood by youngsters when they see that every composite number can be expressed as a product of prime factors that is unique except for the order of the factors.

CANCELLATION LAW AND UNIQUE FACTORIZATION THEOREM FOR STRING OPERATIONS

2006 ◽  
Vol 05 (02) ◽  
pp. 231-243
Author(s):  
DONGVU TONIEN
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Additive Group ◽  
New Product ◽  
Binary String ◽  
Factorization Theorem ◽  
General Criterion ◽  
Unique Factorization ◽  
Product Operation ◽  
Associative Law

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Zm, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved. In this paper, we extend Hoit's results by replacing the Abelian group (Zm, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented.

scholarly journals Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

The Fundamental Theorem of Arithmetic Dissected

1997 ◽  
Vol 81 (490) ◽  
pp. 53 ◽  
Author(s):  
Ahmet G. Agargun ◽  
Colin R. Fletcher
Keyword(s):  
Fundamental Theorem ◽  
Fundamental Theorem Of Arithmetic

Several weighted sum formulas of multiple zeta values

2017 ◽  
Vol 13 (09) ◽  
pp. 2253-2264 ◽  
Author(s):  
Minking Eie ◽  
Wen-Chin Liaw ◽  
Yao Lin Ong
Keyword(s):  
Real Number ◽  
Weighted Sum ◽  
Multiple Zeta Values ◽  
Explicit Evaluation ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Number Formula ◽  
Positive Integers ◽  
Special Case

For a real number [Formula: see text] and positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we evaluate the sum of multiple zeta values [Formula: see text] explicitly in terms of [Formula: see text] and [Formula: see text]. The special case [Formula: see text] gives an evaluation of [Formula: see text]. An explicit evaluation of the multiple zeta-star value [Formula: see text] is also obtained, as well as some applications to evaluation of multiple zeta values with even arguments.

Sign in / Sign up

Export Citation Format

Share Document