**Volume 17** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**The relation between ***π*(*x*) and certain arithmetic functions
*Original research paper. Pages 1—9*
Magdalena Corciovei-Bănescu

Full paper (PDF, 178 Kb) |

Abstract We prove an improvement Rosser–Schoenfeld inequalities, more precisely:

where

*A*_{k}(

*x*) = Σ

_{p ≤ x} p^{k} and

*k* ≥ 0.

** A characterization of modularity in graphs**

*Original research paper. Pages 10—12*

Yilun Shang

Full paper (PDF, 121 Kb) | Abstract

We present alternative expressions for modularity in graphs. Modularity is used as a measure to characterize the community of networks, which is one of the most important features in real-world networks, especially social networks.

** Note on ***φ*, *ψ* and *σ*—functions. Part 3

*Original research paper. Pages 13—14*

Krassimir Atanassov

Full paper (PDF, 96 Kb) | Abstract

Equalities connecting *φ*, *ψ* and *σ*—functions are formulated and proved.

** On a class of infinite sequences with relatively prime numbers and twin prime conjecture**

*Original research paper. Pages 15—17*

Blagoy N. Djokov

Full paper (PDF, 116 Kb) | Abstract

In this paper, a class of infinite sequences with positive integer terms is considered. If the Twin Prime Conjecture, stipulating that there are infinitely many twin prime numbers, is true, then the sequence of all twin prime numbers belongs to the same class. In the present investigation, it is proved that the mentioned class is non-empty and moreover there exists at least one element of this class containing all twin primes.

**On the Diophantine equation ***y*^{n} = *f*(*x*)^{n} + *g*(*x*)

*Original research paper. Pages 18—21*

R. Srikanth and S. Subburam

Full paper (PDF, 131 Kb) | Abstract

In this paper, we simplify the algorithm of Szalay

.

** A note on switching in symmetric ***n*—sigraphs

*Original research paper. Pages 22—25*

P. Siva Kota Reddy, B. Prashanth and Kavita S. Permi

Full paper (PDF, 138 Kb) | Abstract

In this note, we define switching in a different manner and obtained some results on symmetric *n*-sigraphs.

**The structure of even powers in ***Z*_{3}: Critical structural factors that prevent the formation of even—powered triples greater than squares

*Original research paper. Pages 26—30*

J. V. Leyendekkers and A. Shannon

Full paper (PDF, 119 Kb) | Abstract

Integer structure analysis illustrates the critical structural factors which underpin the failure of (*N*^{4m} + *M*^{4m}) ever to equal an equivalent power. The number 3 plays a vital role as integers divisible by 3, when raised to an even power of the form 4*m*, have rows in a table of modular rings which are triangular numbers, whereas other integers raised to the same power have rows which are pentagonal numbers. The substructure within these sequences of pentagonal numbers is order within order, analogous to structure in chaos theory.

** The structure of geometric number sequences**

*Original research paper. Pages 31—37*

J. V. Leyendekkers and A. Shannon

Full paper (PDF, 50 Kb) | Abstract

An integer structure analysis (ISA) of the triangular, tetrahedral, pentagonal and pyramidal numbers is developed. The relationships among the elements and the powers of the elements of these sequences are discussed. In particular, the triangular and pentagonal numbers are directly linked and are structurally important for the formation of triples. The class structure in the modular rings Z_{3} and Z_{4} of some elements of these sequences reinforce previous studies of their properties.

** Some properties and applications of a new arithmetic function in analytic number theory**

*Original research paper. Pages 38—48*

Ramesh Kumar Muthumalai

Full paper (PDF, 189 Kb) | Abstract

We introduce an arithmetic function and study some of its properties analogous to Möbius function in analytic number theory. In addition, we derive some simple expressions to connect infinite series and infinite products through this new arithmetic function. Moreover, we study applications of this new function to derive simple identities on partition of an integer and some special infinite products in terms of multiplicative functions.

** Some properties of unitary addition Cayley graphs**

*Original research paper. Pages 49—59*

Deepa Sinha, Pravin Garg and Anjali Singh

Full paper (PDF, 180 Kb) | Abstract

Let Γ be an abelian group and *B* be a subset of Γ. The addition Cayley graph *G*′ = Cay^{+}(Γ, *B*) is the graph having the vertex set *V* (*G*′) = Γ and the edge set *E*(*G*′) = {*ab* : *a* + *b* ∈ *B*}, where *a*, *b* ∈ Γ. For a positive integer *n* > 1, the unitary addition Cayley graph *G*_{n} is the graph whose vertex set is *Z*_{n}, the integers modulo *n* and if *U*_{n} denotes set of all units of the ring *Z*_{n}, then two vertices a, b are adjacent if and only if *a + b ∈ U*_{n}. The unitary addition Cayley graph *G*_{n} is also defined as, *G*_{n} = Cay^{+}(*Z*_{n}, *U*_{n}). In this paper, we discuss the several properties of unitary addition Cayley graphs and also obtain the characterization of planarity and outerplanarity of unitary addition Cayley graphs.

** Erratum to “Modular rings and the integer 3”**

*Erratum. Page 60*

Erratum (PDF, 10 Kb)

**Volume 17** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4