Proceedings of the Sixth International Conference on Number Theory and Smarandache Notions

---

Number theory is an ancient subject, but we still cannot answer many simplest and most natural questions about the integers. Some old problems have been solved, but more arise. All the research for these ancient or new problems implicated and are still promoting the development of number theory and mathematics. American-Romanian number theorist Florentin Smarandache introduced hundreds of interest sequences and arithmetical functions, and presented many problems and conjectures in his life. In , he published a book named Only problems, Not solutions!. He presented unsolved arithmetical problems and conjectures about these functions and sequences in it.

Already many researchers studied these sequences and functions from his book, and obtained important results. This book, Research on Smarandache Problems in Number Theory Collected papers , contains 41 research papers involving the Smarandache sequences, functions, or problems and conjectures on them. All these papers are original.

Some of them treat the mean value or hybrid mean value of Smarandache type functions, like the famous Smarandache function, Smarandache ceil function, or Smarandache primitive function. Others treat the mean value of some famous number theoretic functions acting on the Smarandache sequences, like k-th root sequence, k-th complement sequence, or factorial part sequence, etc. There are papers that study the convergent property of some infinite series involving the Smarandache type sequences.

Some of these sequences have been first investigated too. In addition, new sequences as additive complement sequences are first studied in several papers of this book. Most authors of these papers are my students. After this chance, I hope they will be more interested in the mysterious integer and number theory! More future papers by my students will focus on the Smarandache notions, such as sequences, functions, constants, numbers, continued fractions, infinite products, series, etc.

For example, the first few values of Sdf n are: In reference [1] and [2], Professor F. Smarandache asked us to study the properties of Sdf n. About this problem, some authors had studied it, and obtained some interesting results, see references []. For example, Maohua Le [4] discussed various problems and conjectures about Sdf n , and obtained some useful results, one of them as follows: That is, we will prove the following: Some preliminary lemmas In this section, we shall give several simple lemmas which are necessary in the proof of our theorem.

They are stated as follows: Proofs of Lemma 2 and Lemma 3 can be found in reference [9]. Proofs of Lemma 5 and Lemma 6 can be found in reference [4]. Proof of the theorem In this section, we will complete the proof of our Theorem. This completes the proof of Theorem. On Smarandache Semigroups A. Examples are provided for justification. Keywords Semigroup, regular element, completely regular element, divisibility, idempotent element.

Introduction Smarandache notions on all algebraic and mathematical structures are interesting to the world of mathematics and researchers. The Smarandache notions in groups and the concept of Smaranadache Semigroups, which are a class of very innovative and conceptually a creative structure, have been introduced in the context of groups and a complete possible study has been taken in [11].

Proceedings of the Sixth International Conference on Number Theory and Smarandache Notions

Padilla Raul intoduced the notion of Smarandache Semigroups in the year in the paper Smarandache Algebraic Structures [6]. In [5], the concept of regularity was first initiatied by J. Neumann for elements of rings. In general theory of semigroups, the regular semigroups were first studied by Thierrin [7] under the name demi-groupes inversifs. The completely regular semigroups were introduced by Clifford [2]. The notions of regular element, completely reagular element of a semigroup are very much useful to characterize Smarandache Semigroups.

In this paper we present characterizations of Smarandache Semigroups. Besides, some more theorems on Smarandache Semigroups, examples are provided for justification. In Section 2 we give some basic definitions from the theory of semigroups See [3] and definition of Smarandache Semigroup See [11]. In Section 3 we present our main characterization of Smarandache Semigroups and examples for justification.

On Smarandache Semigroups 39 If b is a right divisor of a, we say that a is divisible on the right by b. If b is a left divisor of a, we say that a is divible on the left by b. Left unit is defined analogously. An element that is both a right and a left unit of some elemet is called two-sided unit of that element. An element I which is its own two-sided unit is called an Idempotent: A semigroup consisting entirely of regular elements is said to be Regular semigroup. A semigroup consisting entirely of completely regular elements is said to be completely regular.

In [3], the following observations are known: Concepts of regularity and complete regularity coincide for commutaive semigroup. Every idempotent is completely regular. It is its own regular two-sided unit. A regular left unit of an arbitrary element is always an idempotent. No element in a semigroup S may have two regular two-sided units. If an element has regular two-sided unit then it is completely regular.

Proofs of the theorems In this section we give characterizations of Smarandache Semigroups by proving the fol- lowing theorems. A semigroup S is a Smarandache Semigroup if and only if S contains idempotents. The identity element e of G is its own two-sided unit i. Hence, S contains idempotent. Conversely, assume that the semigroup S contains idempotents. Let I be an arbitrary idempotent of the semigroup S. Write GI for the set of all completely regular elements of S for 40 A.

Chandra Sekhar Rao which I is a regular two-sided unit. In view of 2. Now we show that GI is a group under the operation on S. Let g1, g2 be any two elements in GI. Since I is a regular two-sided unit of g1 and g2 we have for some u1, u2, v1, v2 in S. Since, I is a two-sided unit of the element g1g2, I is a regular two-sided unit of g1g2.

Therefore GI is a semigroup with unit I. A semigroup S is a Smarandache semigroup if and only if S contains completely regular elements. On the other hand if the semigroup S contains a completely regular element, say a, then a has an idempotent element I as its regular two-sided unit. In view of the Theorem 3. Hence, S is a Smarandache Semigroup. Let S be a Smarandache Semigroup. The set C of all completely regular elements of S can be expressed as the union of non-intersecting groups. In view of Theorem 3.

Number Theory

Examples In this section we give examples for justification. On Smarandache Semigroups 41 e a b c e e a b c a a e b c b b b c b c c c b c Table 1 Clearly, the operation is commutative. Moreover the idempotent elements are e, c. Using the Table 1, we can easily see that Ge and Gc are groups. The composition table is as follows: Acknowledgements The author would like to express his deep gratitude to Dr. Chandra Sekhar Rao References [1] A. A, 22 , Thierrin, Sur une condition necessaire et suffisante pour qu un semigroupe siot un groupe, Compt.

Paris, , China, 5 , In this paper, some results with respect to Merrifield-Simmons index of zig-zag tree-type hexag- onal systems are shown. Using these results, hexagonal chains and hexagonal spiders with the lower bound of Merrifield-Simmons index are also determined. Keywords Merrifield-Simmons index, zig-zag tree-type hexagonal system, hexagonal spider. Hexagonal systems are of great importance for theoretical chemistry because they are the natural graph representations of benzenoid hydrocarbons [2].

A hexagonal system is a tree-type one if it has no inner vertex. The zig-zag tree-type hexagonal systems are the graph representations of an important subclass of benzenoid molecules. A considerable amount of research in mathematical chemistry has been devoted to hexagonal systems []. In order to describe our results, we need some graph-theoretic notations and terminologies. Our standard reference for any graph theoretical terminology is [1].

Let e and u be an edge and a vertex of G, respectively. Let H be a subset of V G. Undefined concepts and notations of graph theory are referred to []. Two vertices of a graph G are said to be independent if they are not adjacent. Denote i G the number of independent sets of G. In chemical terminology, i G is called the Merrifield- Simmons index.

Clearly, the Merrifield-Simmons index of a graph is larger than that of its proper subgraphs. If the subgraph Bn[V3] is a comb, then Bn is called a helicene chain and denoted by Hn see [11]. Denote by Tn the tree-type hexagonal systems containing n hexagons. Let H be a hexagon of T.

Obviously, H has at most three adjacent hexagons in T ; if H has exactly three adjacent hexagons in T , then H is called a full-hexagon of T ; if H has two adjacent hexagons in T , and, moreover, if its two vertices with degree two are adjacent, then call H a turn-hexagon of T ; and if H has at most one adjacent hexagon in T , then H is called an end-hexagon of T. If any branch of T is a zig-zag chain, then T is called zig-zag tree-type hexagonal system.

Both a zig-zag hexagonal chain and zig-zag hexagonal spider are zig-zag tree-type hexagonal systems with no full-hexagon and only one full-hexagon, respectively. Some useful results Among tree-type hexagonal systems with extremal properties on topological indices, Ln and Zn play important roles.

We list some of them about the Merrifield-Simmons index as follows. Let r and s be two adjacent vertices of B of at least degree two. According to Lemma 2. Keep the notations as Lemma 2. The proof of Theorem 3. Zig-zag tree-type hexagonal systems A graph G is called a zig-zag tree-type hexagonal system if it is a tree-type hexagonal system and any branch of which is zig-zag chain. A graph G is called a spider if it is a tree and contains only one vertex of degree greater than 2. For positive integer n1, n2, n3, we use S n1, n2, n3 to denote a hexagonal spider with three legs of lengths n1, n2 and n3, respectively see [11].

If a hexagonal spider S n1, n2, n3 whose 3 legs are linear chains, then such a graph is called a linear hexagonal spider and denoted by L n1, n2, n3 see [11]. Similarly if each leg of S n1, n2, n3 combining with the central hexagon is a zig-zag chain, then such graph is called a zig-zag hexagonal spider and denoted by Z n1, n2, n3 see [11]. By repeating to apply transformation I on a hexagonal spider S n1, n2, n3 and Zn, and according to Theorem 3. Topics in Current Chemistry, Springer, Berlin, Gutman, Extremal hexagonal chains, J.

Hosoya, Topological index, Bull. Japan, 33 , Tian, Extremal hexagonal chains concerning largest eigenvalue, Sci. A, 44 , Tian, Extremal catacondensed benzenoids, J. China Abstract For any positive integer n, the Smarandache reciprocal function Sc n is defined as the largest positive integer m such that y n! Keywords Smarandache reciprocal function, Pseudo-Smarandache dual function, equation, solution. Introduction and results In reference [1], A. Murthy introduced function Sc n , which is called the Smarandache reciprocal function.

It is defined as the largest positive integer m such that y n! For example, the first few values of Sc n are: Some authors had studied the elementary properties of Sc n , and obtained many inter- esting conclusions. About this function, some authors had studied its properties, and obtained a series of interesting results, see references []. K Majumdar studied this problem, and found several counter-examples to the conjec- ture.

The main purpose of this paper is to study the solvability of the equation 1 , and find its all positive integer solutions. That is, we shall prove the following: The equation 1 has infinite solutions, they are: Some useful lemmas Lemma 1. Proof of the theorem In this section, we shall use the elementary methods to complete the proof of our theorem. Hence, there are no solutions. Therefore, there are no solutions.

From the cases i - iii , we know that the equation 1 has no even positive integer solutions.

Therefore, there are no solutions in this case. Now we discuss in the following cases: So in this case, the equation 1 has no solutions. Now we discuss the solutions in the following several parts: In this case, the equation 1 has no solutions. Thus, the theorem is established. Murthy, Smarandache reciprocal function and an elementary inequality, Smarandache Notions Journal, 11 , No.

That is, the numbers that can be partitioned into two groups such that the second one is three times bigger than the first.

The main purpose of this paper is using the elementary method to study the properties of the Smarandache 3n-digital sequence, and solved a related conjecture. Keywords Smarandache 3n-digital sequence, elementary method, conjecture. In reference [1], Professor F. About this problem, professor Zhang proposed the following: There does not exist any complete square number in the Smarandache 3n- digital sequence a n. In reference [2], Jin Zhang studied this problem, and proved the following conclusions: If positive integer n is a complete square number, then a n is not a complete square number.

Equation 1 has solutions, and part of the solutions can be expressed as follows: Proof of the theorems In this section, we will complete the proof of the theorem. Firstly, we prove Theorem 1.

The aim of this paper is to study a class of U -rpp semigroups, namely, left U -rpp semigroups. After giving some characterizations of left U -rpp semigroups, we establish a structure of this kind of semigroups. Keywords Left U -rpp semigroups, right zero bands, U -left cancellative semigroups. Clearly, regular semigroups and rpp semigroups are all U -rpp semigroups.

In fact, left U -rpp semigroups are U -rpp semigroups whose projections are left central. A rpp semigroup with left central idempotents have been studied by Ren-Shum in [2]. It was proved in [2] that the a rpp semigroup S with left central idempotents is isomorphic to a strong semilattice of left cancellative right stripes. In this paper, we will prove that a semigroup S is a left U -rpp semigroup if and only if S is a semilattice of a direct product of a U -left cancellative monoid and a right zero band; if and only if S is a strong semilattice of a direct product of a U -left cancellative monoid and a right zero band.

For any notation and terminologies not given in this paper, the reader is referred to [4], [5] and [6]. Preliminaries Throughout this paper, S,U is a U -semiabundant semigroup. It follows immediately from definition the following results. Let S,U be a left U -rpp semigroup. Moreover, it can easily verified that the set of projections U of a left U -rpp semigroup S,U forms a right normal band.

Let S,U be a left U-rpp semigroup. Suppose that a, b are two any elements of S,U. Suppose that S,U is a left U -rpp semigroup. Finally, we need the following definition in section 3. It is easy to see that left cancellative semigroups are U -left cancellative semigroups. Structure theorem In this section, we will establish a structure theorem for left U -rpp semigroups. The following statements are equivalent on a semigroup S: Then by Theorem 2. This shows that U is a right normal band.

It follows that U is a band. In fact we have proved that S,U is a left U -rpp semigroup. A structure theorem of left U -rpp semigroups 67 References [1] M. Lawson, Rees matrix semigroups, Proc. Shum, Structure theorems for right pp-semigroups with left central idempotents, General Algebra and Applications, 20 , Lawson, Semigroups and orodered categories.

Algebra, , Fountain, Adequate semigroups, Proc. The main purpose of this paper is using the elementary method to study the properties of the Smarandache 5n-digital sequence, and obtained some usefull conclusions.

Keywords The Smarandache 5n-digital sequence, elementary method, conjecture infinite series, convergence. This sequence was first proposed by professor F. Smarandache, he also asked us to study the proper- ties of 5n-digital sequence. About this problem, it seems that none had studied it yet, at least we have not seen any related papers before. Recently, Professor Zhang Wenpeng proposed the following: I think that this conjecture is interesting, because if it is true, then we shall obtain a deeply properties of the Smarandache 5n-digital sequence.

Additional Information

That is, we shall prove the following conclusions: If positive integer n is a square-free number That is, for any prime p, if p n, then p2 - n , then an is not a complete square number. If positive integer n is a complete square number, then an is not a complete square number. Let z be a real number. Proof of the theorems In this section, we shall use the elementary method to complete the proof of our theorems.

First we prove Theorem 1. From above discussion, we know that there does not exist any positive integer u such that formula 3 hold. This proves Theorem 1. Now we prove Theorem 2. This proves Theorem 2. Now we prove Theorem 3. Then from the definition of the Smarandache 5n-digital sequence we have: On the Smarandache 5n-digital sequence 71 Therefore we have: This proves Theorem 3.

Now we prove Theorem 4. This completes the proof of Theorem 4. Smarandache, Sequences of numbers involving in unsolved problem, Hexis, , Conversely, every amenable partial order on S can be constructed in this way. Keywords Locally inverse semigroup, amenable partial order, inverse transversal. Let S be a regular semigroup with set E S of idempotent elements. Thus, a locally inverse semigroup equipped with the natural partial order is a partially ordered semigroup. Particularly, an inverse semigroup is a partially ordered semigroup under the natural partial order. McAlister introduced and studied amenable partially ordered inverse semigroup in [3].

Blyth and Almeida Santos generalized left amenable partial orders on inverse semigroup to regular semigroup with an inverse transversal in [4]. We recall the following definition. Blyth and Almeida Santos gave a complete description of all amenable partial orders on S and showed the natural partial order on S is the smallest amenable partial order in [5].

In this paper, we will give a new characterization of the amenable partial orders on S. Hence, we immediately have the following lemma.

They proved in [5] that an amenable partial order on S can be constructed by a R-cone P and a L-cone Q, conversely, every amenable partial order on S can be obtained in this way see Theorems 8 and 10 in [5]. Now, we have Definition 2. It follows from Definition 2. Then the following lemma is clear. It follows by Lemma 2. We can obtain from Theorem 2. References [1] Petrich M. Introduction and results Dr. In a triangle 4ABC see Fig. In this paper, we shall generalize this theorem for quadrilateral and pentagon.

Some lemmas To complete the proof of the theorems, we need the following several lemmas. Apply the mathematical induction. Now the Lemma 4 follows from the induction. Proof of the theorems In this section, we prove Theorem 1 and Theorem 2. It is clear that in the pentagon see Fig. The translational hull of an inverse semigroup was studied by Ault in [4]. Fountain and Lawson studied the translational hull of adequate semigroups. And later on, the translational hull of strongly right or left adequate semigroups were further investigated by Ren and Shum.

In this paper, we concentrate on the translational hull of strongly right Ehresmann semigroups. It is proved that the translational hull of a strongly right Ehresmann semigroup is still of the same type. Our results extends the previous results of strongly right adequate semigroups. Keywords Translational hulls, strongly right U -ample semigroups, strongly right Ehresm- ann semigroups.

The concept of translational hull of semigroups and rings was first introduced in by Petrich in [8]. The translational hull of an inverse semigroup was first studied by Ault in [4], and the translational hull of an adequate semigroup was further studied by Fountain and Lawson in [1]. And later on, Ren and Shum investigated in the translational hull of a strongly right or left adequate semigroup in [11].

The translational hull of semigroup plays an important role in the theory of semigroups. In this paper, we will show that the translational hull of an strongly right left Ehresmann semigroup is still of the same type. Preliminaries We first give some basic results and notation from [7]. Let a, b be elements of a semigroup S,U. Then the following statements on S,U are equivalent: Then, by Lemma 2.

Now we have directly from definition the following lemma. Let a, b be elements of a strongly right Ehresmann semigroup S,U. Then the following conditions hold in S,U: Let S,U be a strongly right Ehresmann semigroup, the following statements are equivalent: It is clear that from Lemma 2. Suppose that S,U is a projection balanced semigroup S,U. We only need to prove i since the proof of ii can be obtained similarly. The necessity part of i is clear. Thus the proof is completed. Let S,U be a strongly right Ehresmann semigroup.

It is clear that i implies ii and i implies iii. In fact, we only need to show that iii implies i. Strongly right Ehresmann semigroups In this section, we assume always that S,U is a strongly right Ehresmann semigroup with the set of projections U. Let S,U be a strongly right Ehresmann semigroup with semilattice U of projections.

Let a, b be elements of S,U. Then by Lemma 3. Then, we have the following result. Again by Lemma 2. To do this, we need the following crucial Lemma. Assume that e is any projection from U. Then, by Lemma 3. Thus, it follows by Lemma 2. Since S,U is a strongly right Ehres- mann semigroup and by Lemma 3. We next show that it is right compatible.

Since S,U is a strongly right Ehres- mann semigroup, it follows from Lemma 2. Summarizing above these observations, we can prove the following main theorem. The translational hull of a strongly right Ehresmann semigroup is still a strongly right Ehresmann semigroup. The proof is hence completed. As a direct consequence of Theorem 3. The translational hull of a strongly right U -ample semigroup is still a strongly right U -ample semigroup.

Let S,U be a strongly right U -ample semigroup. Lawson, The translational hull of an adequate semigroup, Semi- group Forum, 32 , Ault, The translational hull of an inverse semigroup, Glasgow Math. Shum, On translational hulls of a type-A semigroup, Journal of Algebra, , Petrich, The translational hull in semigroups and rings, Semigroup Forum, 1 , Guo, The translational hull of a strongly right type-A semigroup, Science in China, 43 , Shum, The translational hull of a strongly right or left adequate semigroup, Vietnam Journal of Mathematics, 34 , Shum, On U -orthodox semigroups, Science in China, 52 , Howie, An introduction to semigroup theory, Academic Press, London, Introduction In this paper [1], Vasantha Kandasamy introduced the concept of Smarandache non- associative rings, which we shortly denote as SNA-rings.

This concept derived from the general definition of a Smarandache Structure i. The only non-associative structure found in Smaran- dache algebraic notions are Smarandache groupiods and Smarandache loops introduced in [2] and [3], which are algebraic structures with only a single binary operation defined on them that is non-associative. But SNA-rings are non-associative structures on which are defined two binary operations one associative and other being non-associative and addition distributes over multiplication both from the right and left.

By [1], it is well know that the loop ring is always a SNA-ring, and the groupiod ring is also a SAN-ring when it satisfies some conditions. Those results motivate us to find the smallest non-associative ring By smallest we mean the number of elements in them that is order is the least that is we can not find any other non-associative ring of lesser order than that. In this note, we shall give some interesting results about the mentioned problems in [1].

A set S together with a binary operation is a groupoid. In view of these we have the following interesting results. Let R be a non-associative ring, which is a Bol ring.

• Leadership Wisdom: Discovering the Lessons of Experience (J-B CCL (Center for Creative Leadership))
• Powers: A Study in Metaphysics
• Divertimento No.11 D Major KV251 - Score
• Not-for-Profit Accounting, Tax, and Reporting Requirements
• Daughter of the Drow: Starlight & Shadows, Book I
• In the Joy of Surrender: Messages of Love from God
• Press Censorship in Jacobean England