smarandache notions journal, vol. 11

8/14/2019 Smarandache Notions Journal, Vol. 11

1/327

Vol. 11, No. 1-2-3, Spring 2000 ISSN 1084-2810

SMARANDACHE NOTIONS JOURNAL

Mathematical Associationof the

UNIVERSITY OF CRAIOVA


2/327

The table on the first cover represents a Smarandache Group underthe operation /+

4\ [Talukdar, 19].

This journal is an extension of the "Smarandache FunctionJournal", ISBN 1053-4792, Vol. 1-6, 1990-1995.

SNJ is yearly published (in the Spring - and sometimes in theFall) in a 250-350 pages volume, and 800-1000 copies.

SNJ is a referred journal.

SNJ is reviewed, indexed, citated by the following journals:"Zentralblatt F hr Mathematik" (Germany), "Referativnyi Zhurnal"and "Matematika" (Academia Nauk, Russia), "Mathematical Reviews"(USA), "Computing Review" (USA), "Libertas Mathematica" (USA),"Octogon" (Romania), "Indian Science Abstracts" (India), "Ulrich'sInternational Periodicals Directory" (USA), "Gale Directory ofPublications and Broadcast Media" (USA), "Historia Mathematica"(USA), "Journal of Recreational Mathematics" (USA), "The

Mathematical Gazette" (U.K.), "Abstracts of Papers Presented tothe American Mathematical Society" (USA), "Personal ComputerWorld" (U.K.), "Mathematical Spectrum" (U.K.), "Bulletin of Pureand Applied Sciences" (India), Institute for ScientificInformation (PA, USA), "Library of Congress Subject Headings"(USA).

SNJ is catalogued by the Library of Congress, Washington, D. C.,at Call# QA246 .S63.

Copyright 2000by Mathematical Association of the University ofCraiova, 13 Al. I. Cuza Street, R-1100, Romania.

Referents :Y. V. Chebrakov , Department of Mathematics,

St.-Petersburg Technical University,Nevsky 3-11, 191186, St.-Petersburg, Russia.

Ion Goian , Department of Algebra, Number Theory, and Logic,State University of Kishinev, Republic of Moldova.

Florian Luca , Mathematical Institute, Czech Academy ofSciences, Zitna 25, 11567 Praha 1, Czech Republic.

Vasile Marin , Department of Algebra, Number Theory, and Logic,State University of Kishinev, Republic of Moldova.

Amarnath Murthy , S. E. (E&T), Well Logging Services, Oil And

Natural Gas Corporation Ltd., Sabarmati, Ahmedbad, India -380005.

Sebastian Martin Ruiz , Avda. De Regla, 43,Chipiona 11550 (Cadiz), Spain.

Dr. Felice Russo, Via A. Infante 7, 67051 Avezzano (Aq), Italy.


3/327

Sabin T | brc | , Department of Computer Science, University, Bra ov, Romania.

Dwiraj Talukdar, Head Department of Mathematics, NalbariCollege, Nalbari, Assam, India.

ASSOCIATE EDITORIAL BOARD:

Charles Ashbacher , Charles Ashbacher Technologies, Box 294,119 Northwood Drive, Hiawatha, Iowa 52233, USA.

Krassimir Atanassov , Bulgarian Academy of Sciences, P. O. Box12 MRL, Sofia 1113, Bulgaria.Robert Backstrom , Post Office 162, Gymea, New Wales,

Australia. Mihaly Bencze , Str. H | rmanului 6, 2212 S | cele 3, Jud. Bra ov,

Romania.Emil Burton , Department of Mathematics, University of Cluj-

Napoca, Romania.Ion Cojocaru & Sorin Cojocaru , Department of Mathematics,

University of Craiova, Romania.Constantin Corduneanu, Department of Mathematics, University

of Texas at Arlington, USA. Micha Fleuren , Acaciastraat 16, 6598 BC Heyen, The

Netherlands.Patrick De Geest, Oppemlaan 141, B-1970 Wezembeek-Oppem,

Belgium.Henry Ibstedt , Glimminge 2036, 280 60 Broby, Sweden.

Maohua Le, Department of Mathematics, Zhanjiang NormalCollege, 29 Cunjin Road, Chikan, Zhanjiang, Guangdong, P. R.China.

A.A.K. Majumdar, Department of Mathematics, JahangirnagarUniversity, Savar, Dhaka 1342, Bangladesh.

Mike Mudge , 22 Gors Fach, Pwll-Trap, St Clears,Carmarthenshire, Dyfed SA33 4AQ, United Kingdom.

Jean Normand , 3, square Robinson, 92330 Sceaux, France.Stefan Porubsky , Department of Mathematics, Institute of

Chemical Technology, Technicka 1905, 16628 Prague 6, CzechRepublic.

E. R | descu & N. R | descu , Department of Mathematics,University of Craiova, Romania.

Carmen Roc oreanu , Department of Mathematics, University ofCraiova, Romania.

Francisco Bellot Rosado , Dos de Mayo 16-8 E dcha, E-47004 Valladolid, Spain.

Jozsef Sandor , Department of Mathematics, Babe -BolyaiUniversity, 3400 Cluj-Napoca, Romania.

K. R. S. Sastry , Jeevan Sandhya, Doddakalsandra Post,Raghuvana Halli, Bangalore 560 062, India.

Y. A. Shmagin , Department of Mathematics, St.-PetersburgTechnical University, Nevsky 3-11, 191186,


4/327

St.-Petersburg, Russia.Vera W. de Spinadel , Centro de Matematica y Diseno MAyDI,

Facultad de Arquitectura, Diseno y Urbanismo, Universidadde Buenos Aires, Argentina.

Homer B. Tilton , Department of Mathematics, Phisics, and Astronomy, Pima Community College, Tucson, Arizona, USA.

INDICATIONS TO AUTHORS

Authors of papers concerning any of Smarandache type functions,numbers, sequences, integer algorithms, paradoxes, experimentalnon-euclidean geometries, algebraic structures, neutrosophic

probability/statistics/set/logic, quantum physics, etc. areencouraged to submit manuscripts to the Editors:

V. Seleacu & I. B | l | cenoiuDepartment of MathematicsUniversity of Craiova, Romania.

The submitted manuscripts may be in the format of remarks,conjectures, solved/unsolved or open new proposed problems, notes,articles, miscellaneous, etc. They must be original work andcamera ready [typewritten/computerized, format: 8.5 x 11 inches ( . 21,6 x 28 cm)]. They are not returned, hence we advise theauthors to keep a copy.

The title of the paper should be writting with capital letters.The author's name has to apply in the middle of the line, near

the title. References should be mentioned in the text by a numberin square brackets and should be listed alphabetically. Currentaddress followed by e-mail address should apply at the end of the

paper, after the references.The paper should have at the beginning an up to a half-page

abstract, followed by the key words. All manuscripts are subject to anonymous review by two or more

independent sources.Every letter will be answered.The authors will receive free copies of the journal.

Papers in electronic form are accepted. They can be e-mailed in Microsoft Word 97 (or lower) for Windows 98, WordPerfect 6.0 (or

lower) for Windows 95, HTML, or text files.

Books on Smarandache notions can be ordered from:Bell & Howell Information and Learning(former University of Microfilm International)300 North Zeeb RoadP.O. Box 1346Ann Arbor, Michigan 48106-1346, USATel.: 1-800-521-0600 ext. 4806.

In year 2000 our journal celebrates 10 years of existence.


5/327


6/327


7/327


8/327


9/327


10/327


11/327


12/327


13/327


14/327


15/327


16/327


17/327


18/327


19/327


20/327


21/327


22/327


23/327


24/327


25/327


26/327


27/327


28/327


29/327


30/327


31/327


32/327


33/327


34/327


35/327


36/327


37/327


38/327


39/327


40/327


41/327


42/327


43/327


44/327


45/327


46/327


47/327


48/327


49/327


50/327


51/327


52/327


53/327


54/327


55/327


56/327


57/327


58/327


59/327


60/327


61/327


62/327


63/327


64/327


65/327


66/327


67/327


68/327


69/327


70/327


71/327


72/327


73/327


74/327


75/327


76/327


77/327


78/327


79/327


80/327


81/327


82/327


83/327


84/327


85/327


86/327


87/327


88/327


89/327


90/327


91/327


92/327


93/327


94/327


95/327


96/327


97/327


98/327


99/327


100/327


101/327


102/327


103/327


104/327


105/327


106/327


107/327


108/327


109/327


110/327


111/327


112/327


113/327


114/327


115/327


116/327


117/327


118/327


119/327


120/327


121/327


122/327


123/327


124/327


125/327


126/327


127/327


128/327


129/327


130/327


131/327


132/327


133/327


134/327


135/327


136/327


137/327


138/327


139/327


140/327


141/327


142/327


143/327


144/327


145/327


146/327


147/327


148/327


149/327


150/327


151/327


152/327


153/327


154/327


155/327


156/327


157/327


158/327


159/327


160/327


161/327


162/327


163/327


164/327


165/327


166/327


167/327


168/327


169/327


170/327


171/327


172/327


173/327


174/327


175/327


176/327


177/327


178/327


179/327


180/327


181/327


182/327


183/327


184/327


185/327


186/327


187/327


188/327


189/327


190/327


191/327


192/327


193/327


194/327


195/327


196/327


197/327


198/327


199/327


200/327


201/327


202/327


203/327


204/327


205/327


206/327


207/327


208/327


209/327


210/327


211/327


212/327


213/327


214/327


215/327


216/327


217/327


218/327


219/327


220/327


221/327


222/327


223/327


224/327


225/327


226/327


227/327


228/327


229/327


230/327


231/327


232/327


233/327


234/327


235/327


236/327


237/327


238/327


239/327


240/327


241/327


242/327


243/327


244/327


245/327


246/327


247/327


248/327


249/327


250/327


251/327


252/327


253/327


254/327


255/327


256/327


257/327


258/327


259/327


260/327


261/327


262/327


263/327


264/327


265/327


266/327


267/327


268/327


269/327


270/327


271/327


272/327


273/327


274/327


275/327


276/327


277/327


278/327


279/327


280/327


281/327


282/327


283/327


284/327


285/327


286/327


287/327


288/327


289/327


290/327


291/327


292/327


293/327


294/327


295/327


296/327


297/327


298/327


299/327


300/327


301/327


302/327


303/327


304/327


305/327


306/327


307/327


308/327


309/327


310/327


311/327


312/327


313/327


314/327


315/327


316/327


317/327


318/327


319/327


320/327


321/327


322/327


323/327

C O N T E N T S

ARTICLES:DWIRAJ TALUKDAR , D-Form Of Smarandache Groupoid . . . . . . . . 5DWIRAJ TAKUKDAR , The Notions Of The Smarandache Group And The

Smarandache Boolean Ring . . . . . . . . . . . . . . . . . 16 ANGELA VASIU , NICOLAIE OPREA , From Bolyais Geometry To

Smarandache Anti-Geometry . . . . . . . . . . . . . . . . . 24 ANGELA VASIU , Smarandaches New Geometries: A Provocation For An

Ammelioration Of Human Condition . . . . . . . . . . . . . . 29SABIN TABIRCA , TATIANA TABIRCA , The Average Of The Erdos Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31SABIN TABIRCA , TATIANA TABIRCA , On The Convergence Of The Erdos

Harmonic Series . . . . . . . . . . . . . . . . . . . . . . 34


324/327

FELICE RUSSO , An Experimental Evidence On The Validity Of Third Smarandache Conjecture On Primes . . . . . . . . . . . . . 38

FELICE RUSSO , Some Results About Four Smarandache U-ProductSequences . . . . . . . . . . . . . . . . . . . . . . . . . 42

FELICE RUSSO , On An Unsolved Question About The SmarandacheSquare- Partial-Digital Subsequences . . . . . . . . . . . . 50CHARLES ASHBACHER , On A Conjecture By Russo . . . .. . . . . . 53SEBASTIAN MARTIN RUIZ , A Functional Recurrence To Obtain The Prime

Numbers Using The Smarandache Coprime Function . . .. . . . . 56SEBASTIAN MARTIN RUIZ , The General Term Of The Prime Number

Sequence And The Smarandache Prime Function . . . . . . . . 59SEBASTIAN MARTIN RUIZ , Expressions Of The Smarandache Coprime

Function . . . . . . . . . . . . . . . . . . . . . . . . . . 62 MLADEN V. VASSILEV-MISSANA , KRASSIMIR ATANASSOV , Note On the

Diophantine Equation 2x 2 -3y 2 =p . . . . . . . . . . . . . . . 64HRISTO ALADJOV , KRASSIMIR ATANASSOV , Remark On The 62-th

Smarandache Problem . . . . . . . . . . . . . . . . . . . . 69 MAOHUA LE , The Integral Values Of log S(n k ) . . . . . . . . . 71

k nRONGI CHEN , MAOHUA LE , On The Functional Equation S(n) 2 +S(n)=kn 73ZHONG LI , MAOHUA LE , On The Functional Equation Z(n)+ (n)=d(n) 77

M. BENCZE , Smarandache Relationships And Subsequences . . . . 79SYLVESTER SMITH , A Set Of Conjectures On Smarandache Sequences 86

FANEL IACOBESCU , Smarandache Partition Type And Other Sequences 93HELEN MARIMUTHA , Smarandache Concatenate Type Sequences . . . 97 MIHALY BENCZE , Smarandache Recurrence Type Sequences . . . . . 99

SABIN TABIRCA , About Smarandache-Multiplicative Functions . . 103 ANGHEL N. RUGINA , On Smarandache Mixed Non-Euclidean

Geometries . . . . . . . . . . . . . . . . . . . . . . . . 105 K. R. S. SASTRY , Smarandache Number Related Triangles . . . . 107CHARLES ASHBACHER , Solutions To Some Sastry Problems On

Smarandache Number Related Triangles . . . . . . . . . . . 110 MAOHUA LE , On The Difference S(Z(n))-Z(S(n)) . . . . . . . . 116 ANTHONY BEGAY , Smarandache Ceil Functions . . . . . . . . . . 117

CHARLES ASHBACHER , Some Problems Concerning The SmarandacheDeconstructive Sequence . . . . . . . . . . . . . . . . . . 120

EMIL BURTON , S-Primality Degree Of A Number And S-PrimeNumbers . . . . . . . . . . . . . . . . . . . .. . . . . . . . 123E. RADESCU , Some Elementary Algebraic Considerations Inspired By

Smarandache Type Functions (II) . . . . . . . . . . . . . . 124RONGJI CHEN , On The Functional Equation (S(n))

r + (S(n))

r-1+ +

S(n) = n . . . . . . . . . . . . . . . . . . . . . . . . . 128ZHONG LI , Solution Of Two Questions Concerning The Divisor Function And The Pseudo-Smarandache Function . . . . . . . 132J. SANDOR , On A Conjecture Of Smarandache On Prime Numbers . 136FLORIAN LUCA , On The Smarandache Irrationality Conjecture . . 137J. SANDOR , A Note On S(n), Where n Is An Even Perfect Number 139STEVEN FINCH , Moments Of The Smarandache Function . . . . . . 140GHEORGHE DINULESCU , The Intangible Absolute Truth . . . . . .. 142Y. V. CHEBRAKOV , Advance Of Smarandache Approach To Solving Systems Of Diophantine Equations . . . . . . . . . . . . . . 144Y. V. CHEBRAKOV , V. V. SHMAGIN , Analytical Approach To Description


325/327

, , a yt ca pp oac o esc pt oOf Some Combinatorial And Number-Theoretic Computative

Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 156I. BALACENOIU , D. BORDEA , V. SELEACU , Numerical Functions And

Triplets . . . . . . . . . . . . . . . . . . . . . . . . . 165 AMARNATH MURTHY , Explorating Some New Ideas On Smarandache type

Sets, Functions And Sequences . . . . . . . . . . . . . . . 171 AMARNATH MURTHY , Some New Smarandache Sequences, Functions And

Partitions . . . . . . . . . . . . . . . . . . . . . . . . 179 AMARNATH MURTHY , On The Divisors Of Smarandache Unary Sequence 184

CHARLES ASHBACHER , On Iterations That Alternate The Pseudo-Smarandache And Classic Functions Of Number Theory . . . . 186

FLORIAN MUNTEANU , OCTAVIAN MUSTAFA , The Almost PresumableMaximality Of Some Topological Lemma . . . . . . . . . . . 192

SABIN TABIRCA , TATIANA TABIRCA , On The Convergence Of The Euler Harmonic Series . . . . . . . . . . . . . . . . . . . . . . 196

J. SANDOR , On Two Notes By M. Bencze . . . . . . . . . . . . 201J. SANDOR , On Certain Generalizations Of The Smarandache Function

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 VASILE SELEACU , On The Numerical Function S min -1 . . . . . . . 213

CHARLES ASHBACHER , On Numbers Where The Values Of The Pseudo-Smarandache Function Of It And The Reversal Are Identical . 215

AMARNATH MURTHY , Smarandache Reciprocal Partition Of Unity SetsAnd Sequences . . . . . .. . . . . . . . . . . . . . . . . 218

AMARNATH MURTHY, Generalization Of Partition Function,Introducing Smarandache Factor Partition . . . . . .. . 227

AMARNATH MURTHY , A General Result On The Smarandache Star Function . . . . . . . . . . . . . . . . . . . . . . . . . 240

AMARNATH MURTHY , More Results And Applications Of TheGeneralized Smarandache Star Function . . . . . . . . . . 253

AMARNATH MURTHY , Properties Of Smarandache Star Triangle . . 263 AMARNATH MURTHY , Smarandache Factor Partitions Of A Typical

Canonical Form . . . . . . . . . . . . . . . . . . . . . . 270

AMARNATH MURTHY , Length/Extent Of Smarandache Factor Partitions . . . . . . . . . . . . . . . . . . . . . . . 275

AMARNATH MURTHY , Some More Ideas On Smarandache Factor Partitions . . . . . . . . . . . . . . . . . . . . . . . 280

AMARNATH MURTHY , A Note On The Smarandache Divisor Sequences 284 AMARNATH MURTHY , Algorithm For Listing Of Smarandache Factor

Partitions . . . . . . . . . . . . . . . . . . . . . . . . 286 AMARNATH MURTHY , Expansion Of x n In Smarandache Terms Of

Permutations . . . . . . . . . . . . . . . . . . . . . . . 290 AMARNATH MURTHY , Miscellaneous Results And Theorems On

Smarandache Terms And Factor Partitions . . . . . . . . . 297 AMARNATH MURTHY , Smarandache Maximum Reciprocal Representation

Function . . . . . . . . . . . . . . . . . . . . . . . . . 305 AMARNATH MURTHY , Open Problems And Conjectures On The

Factor/Reciprocal Partition Theory . . . . . . . . . . . . 308 AMARNATH MURTHY , Smarandache Reciprocal Function And An

Elementary Inequaliy . . . . . . . . . . . . . . . . . . . 312 AMARNATH MURTHY , Some Funny Examples Of Smarandache Lucky

Methods In Algebra, Trigonometry, And Calculus . . . . . . 316


326/327

REVIEWS:Book Reviews ( CHARLES ASHBACHER , TONNY SHANNON ) . . . . . . 321


327/327

smarandache notions journal, vol. 11

Documents