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..., LIU DUANSEN (EDITORS) RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY (VOL. II) PROCEEDINGS OF THE FIRST NORTHWEST CONFERENCE ON NUMB... ...ORY (VOL. II) PROCEEDINGS OF THE FIRST NORTHWEST CONFERENCE ON NUMBER THEORY Hexis Phoenix, AZ 2005 EDITORS: ZHANG WEN... ...UO, SHAANXI, P.R.CHINA RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY (VOL. II) PROCEEDINGS OF THE FIRST NORTHWEST CONFERENCE ON NUMB... ...ORY (VOL. II) PROCEEDINGS OF THE FIRST NORTHWEST CONFERENCE ON NUMBER THEORY Hexis Phoenix, AZ 2005 ... ... Hexis Phoenix, AZ 2005 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest ... ...Florentin Smarandache, who listed many new and unsolved problems in number theory. Preface Arithmetic is where numbers run across your mind lookin... ...(n)wekonwthat δ k (n) and b m (n) both are multiplicative functions. So in order to findA, we only discuss the problem in casen = p α . Ifn = p α and (...

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...imply a 1 =a 2 =1, a 2k+1 =a k+1 -1, a 2k+2 =a k+1 +1 This book can be ordered in paperbound form from: Books on Demand ProQuest Information... ...sting iteration question was posed in the book Unsolved Questions in Number Theory (first edition) by R.K. Guy. Why does the repetitive application ... ...such that 1+2+…+N is divisible by n. 3) The Smarandache Ceil Function of order k, S k (n): S k (n)=N where N is he smallest positive integer for ... ...ize n+1 and arrange the factors (α 1 +1), (α 2 +1) … (α j +1) in descending order. Let us first assume that α 1 >α 2 > … >α j then, remembering tha... ...5 3 N α α α α = where the primes are assigned to the exponents in ascending order starting with p 1 =3. Every factor in (10) corresponds to a differ... ... k at the lower end and Z(2 k )=2 k+1 -1 at the upper end of the cycle. In order to have a 2-cycle we must find a solution to the equation φ(2 k... ...tions That Alternate the Pseudo-Smarandache and Classic Functions of Number Theory, Smarandache Notions Journal, Vol. 11, No 1-2-3. 26 ... ...tions That Alternate the Pseudo-Smarandache and Classic Functions of Number Theory, Smarandache Notions Journal, Vol. 11, No 1-2-3. 4. G.H. Hardy ... ... Vol. 11, No 1-2-3. 4. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers. Oxford University Press, 1938. ...

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...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 impl...

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...RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY (Collected papers) Edited by ZHANG WENPENG D... ... Hexis 200 4 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY (Collect ed papers) Edited by ZHANG W... ...University Xi’an, P. R. China Hexis 2004 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest ... ...Florentin Smarandache, who listed many new and unsolved problems in number theory. Contents Dedication v Preface xi An arithmetic function and the pr... ...its mean value 49 Gao Nan viii RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY On the generalized constructive set 53 Gou su On the inferior and su... ...Moving the integral line from to . This time, the function has a second order pole point at with residue An arithmetic function and the divisor p... ... we move the integral line from to . This time, the function has a three order pole point at with residue where are the computable constants. Note... ... main term we move the integer line from to , then the function have one order pole point at with residue where are computable constants. So we ca... ... obtain the asymptotic formula if . If , then the function have a second order pole point at . Let Res denotes its residue, so we have Noting that...

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...Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences Editor’s Note This book ar... ...OWA 42402 USA GENERALIZED PARTITIONS AND SOME NEW IDEAS ON NUMBER THEORY AND SMARANDACHE SEQUENCES Hexis P... ... Hexis Phoenix 2005 2 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest ... ...e On Maohua Le’s Proof of Murthy’s Conjecture On Reciprocal Partition Theory ………………. 18 1.4 Generalization of Partition Functions, Introducti... ...9 1.5 Open Problems and Conjectures On the Factor/Reciprocal Partition Theory ………………. 28 1.6 A General Result on the Smarandache Star Function... ... r=1 f RP (n) = order of the set SRRPS(n). For example, SRRPS(1) = { (1) }, f RP (1)... ... r=1 f DP (n) = order of SDRPS(n). For example: SRRPS(1) = { (1) }, f RP (1) = 1. SR... ...s. Two partitions are not considered different if they differ only in the order of their summands. Many results concerning the partition function we... ... SFP and factors that F( α 1 , α 2 ) = F( α 2 , α 1 ) and in general the order of the α i in ( α 1 , α 2 , . . . , α r ) is immaterial. Also, the ...

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...se the process may lead to an invariant. The second problem with the initial article is that, say, 7 is considered as 07 or 007 as the case may be in order make its reverse to be 70 or 700. However, the reverse of 7 is 7. In order not to loose the beauty of these sequences the author has introduced stringent definitions to prevent the sequences from collapse when the rever...

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...y. This book assumes that the reader should have a good background of algebraic structures like semigroup, group etc. and a good foundation in number theory....

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...3 3 3 0 3 e 3 0 4 4 2 5 2 5 e 5 5 5 1 4 1 4 1 e 2 This book can be ordered in a paper bound reprint from: Books on Demand ProQu... ...lgebraic structures like semigroup, group etc. and a good foundation in number theory. In Chapter 1 we just recall the basic notations and some imp... ...an identity element e in G if a ∗ e = e ∗ a = a for all a ∈ G. We call the order of the groupoid G to be the number of distinct elements in it de... .... If the number of elements in G is finite we say the groupoid G is of finite order or a finite groupoid otherwise we say G is an infinite groupoid. ... ...ly (G, ∗) is a non commutative groupoid and does not contain an identity. The order of this groupoid is 5. Example 1.2.2: Let (S, ∗) be a groupoid... ... 3 x 2 x 2 x 2 x 1 x 3 x 3 x 3 x 2 x 1 This is a groupoid of order 3, which is non associative and non commutative. Example 1.2.3... ...nger Verlag, (1958). 2. Ivan Nivan and H.S.Zukerman, Introduction to number theory, Wiley Eastern Limited, (1989). 3. W.B. Vasantha Kandasamy, ... ...d Kama. Her current interests include Smarandache algebraic structures, fuzzy theory, coding/ communication theory. In the past decade she has guide... ...cholars in the different fields of non-associative algebras, algebraic coding theory, transportation theory, fuzzy groups, and applications of fuzzy...

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...This book contains short notes or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geometric theorems; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals essentially ...

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... American Research Press Rehoboth 2002 This book can be ordered in microfilm format from: Books on Demand ... ...s or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geometric the- orems; Di... .... . . . . . . . . . . . . 241 8 On certain new conjectures in prime number theory . . . . . . . . . . . . . 243 9 On certain conjectures by Russo . . ... ...ve made it clear that there exists a very intimate correlation between the Theory of numbers and other departments of Mathematics, not excluding geome... ...or simply, H-triangle) if its area = Area(ABC) is an integer number. The theory of H-triangles has a long history and certain results are many times... ...ying (xy 4)(z +t) 2 8(x +y)(z +t) (xy + 4)(x +y) 2 0 (4) This second order inequality gives at once z +t 4(x +y) + p 16(x +y) 2 + (x +y) 2 (xy ... ...n positive integers occurs many times for particular values of a;b;c;d. In order to obtain all solutions of (1), remark that (1) can be written as bdx... ...ve bd = 8 so Yj8, i.e. Y 0 2 f1; 2; 4; 8g. Then Z 0 2f8; 4; 2; 1g (in this order!). By ad = 4, by analyzing (3k 8)j4, (3k 4)j4, (3k 2)j4, (3k 1)j4... ... of [2] asks for the determination of all matrices A2 M 2 [Z] (i.e. second order matrices with integer coe cients) such that A 2 = E, whereE is the u...

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...e Sequences, Stereograms and Series Hexis Phoenix 2005 2 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information... ...0 . . . 0987654321. Some additional runs of the program were performed in order to examine some additional properties of the sequence. The following ... ...e Fibonacci and Lucas sequences. References 1. Unsolved Problems in Number Theory, edited by Richard K. Guy, Springer, 2004, ISBN 0387208607. 70 Chap... ...t we will use in our plots. Points in the quarter plane will be defined as ordered pairs of integers (m, n) and we will iterate (S(m),S(n)) until both... ...integers (m, n) and we will iterate (S(m),S(n)) until both elements of the ordered pair are fixed points. The number of iterations until a double fixe... ...hat kind of stereograms can be created by iterating combinations of number theory functions and plotting the number of iterations? 3.2 Altering the Ba... ...t primes, i. e. n = p 1 p 2 . . . p k , where the primes are in ascending order. Then S(n) = p k and S(n)! contains all of the other primes and comp...

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...totic properties, series convergence, solutions about special equations. At the same time, we put forward to some new interesting problems either in order to research further. We hope this booklet will guide and inspire readers to these fields....

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...是数学中的皇冠”. 虽然数论中的许多问题在很早就开始了研究, 并得到了丰硕的成果, 但是至今仍有许多被数学家称之为“皇冠上的明 珠”的悬而未解的问题等待人们去解决. 正因如此, 数论才能不断地充 实和发展, 才能既古老又年轻, 才能始终活跃在数学领域的前沿. Foreword Number theory, this discipline was originally started from the study integer, so called integer number theory. Later integer further development of number theory, number theory called up. Rather, number theory is an integer of study...

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... (Romania), and Archives of State (Rm. Valcea, Romania). The Smarandache's universe is undoubtedly very fascinating and is halfway between the number theory and the recreational mathematics. Even though sometime this universe has a very simple structure from number theory standpoint, it doesn't cease to be deeply mysterious and interesting. This book, following the Smarand...

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...A SET OF NEW SMARANDACHE FUNCTIONS, SEQUENCES AND CONJECTURES IN NUMBER THEORY Felice Russo AMERICAN RESEARCH PRESS Lupton USA 2000 Wrongness of n ... ...universe is undoubtedly very fascinating and is halfway between the number theory and the recreational mathematics. Even though sometime this univers... ...Even though sometime this universe has a very simple structure from number theory standpoint, it doesn't cease to be deeply mysterious and interesting... ...lems, new Smarandache type sequences and new Smarandache Notions in number theory. Moreover a chapter (IV) is dedicated to the analysis of Smarandach... ...the analysis of Smarandache Double factorial function introduced in number theory by F. Smarandache ("The Florentin Smarandache papers" special colle... ...Q numbers see [1]) ? • Within the chaotic behaviour is there some signs of order ? (namely this numbers exhibit approximate Period Doubling, Self-Simi... ...]. Smarandache conjectured that there are no Smarandache Wrong numbers. In order to check the validity of this conjecture up to some value 0 N , an U...