Smarandache Notions Journal, Vol. 11

Vol. 11, No. 1-2-3, Spring 2000

ISSN 1084-2810

SMARANDACHE NOTIONS JOURNAL

Mathematical Association of the

UNIVERSITY OF CRAIOVA

The table on the first cover represents a Smarandache Group under the operation /+4\ [Talukdar, 19]. This journal is an extension of the "Smarandache Journal", ISBN 1053-4792, Vol. 1-6, 1990-1995.

Function

SNJ is yearly published (in the Spring - and sometimes in the Fall) 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 Fhr 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's International Periodicals Directory" (USA), "Gale Directory of Publications and Broadcast Media" (USA), "Historia Mathematica" (USA), "Journal of Recreational Mathematics" (USA), "The Mathematical Gazette" (U.K.), "Abstracts of Papers Presented to the American Mathematical Society" (USA), "Personal Computer World" (U.K.), "Mathematical Spectrum" (U.K.), "Bulletin of Pure and Applied Sciences" (India), Institute for Scientific Information (PA, USA), "Library of Congress Subject Headings" (USA). SNJ is catalogued by the Library of Congress, Washington, D. C., at Call# QA246 .S63. Copyright 2000 by Mathematical Association of the University of Craiova, 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 of Sciences, 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. 2

Sabin T|bârc|, Department of Computer Science, University, BraÕov, Romania. Dwiraj Talukdar, Head Department of Mathematics, College, Nalbari, Assam, India.

Nalbari

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. Box 12 MRL, Sofia 1113, Bulgaria. Robert Backstrom, Post Office 162, Gymea, New Wales, Australia. Mihaly Bencze, Str. H|rmanului 6, 2212 S|cele 3, Jud. BraÕov, Romania. Emil Burton, Department of Mathematics, University of ClujNapoca, 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 Normal College, 29 Cunjin Road, Chikan, Zhanjiang, Guangdong, P. R. China. A.A.K. Majumdar, Department of Mathematics, Jahangirnagar University, 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, Czech Republic. Department of Mathematics, E. R|descu & N. R|descu, University of Craiova, Romania. Carmen RocÕoreanu, Department of Mathematics, University of Craiova, Romania. Francisco Bellot Rosado, Dos de Mayo 16-8E dcha, E-47004 Valladolid, Spain. Jozsef Sandor, Department of Mathematics, BabeÕ-Bolyai University, 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.-Petersburg Technical University, Nevsky 3-11, 191186, 3

St.-Petersburg, Russia. Vera W. de Spinadel, Centro de Matematica y Diseno MAyDI, Facultad de Arquitectura, Diseno y Urbanismo, Universidad de 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, experimental non-euclidean geometries, algebraic structures, neutrosophic probability/statistics/set/logic, quantum physics, etc. are encouraged to submit manuscripts to the Editors: V. Seleacu & I. B|l|cenoiu Department of Mathematics University 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 and camera ready [typewritten/computerized, format: 8.5 x 11 inches (. 21,6 x 28 cm)]. They are not returned, hence we advise the authors 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 number in square brackets and should be listed alphabetically. Current address 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 Road P.O. Box 1346 Ann Arbor, Michigan 48106-1346, USA Tel.: 1-800-521-0600 ext. 4806. In year 2000 our journal celebrates 10 years of existence. 4

C O N T E N T S ARTICLES: DWIRAJ TALUKDAR, D-Form Of Smarandache Groupoid . . . . . . . . 5 DWIRAJ TAKUKDAR, The Notions Of The Smarandache Group And The Smarandache Boolean Ring . . . . . . . . . . . . . . . . . 16 ANGELA VASIU, NICOLAIE OPREA, From Bolyai’s Geometry To Smarandache Anti-Geometry . . . . . . . . . . . . . . . . . 24 ANGELA VASIU, Smarandache’s New Geometries: A Provocation For An Ammelioration Of Human Condition . . . . . . . . . . . . . . 29 SABIN TABIRCA, TATIANA TABIRCA, The Average Of The Erdos Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 SABIN TABIRCA, TATIANA TABIRCA, On The Convergence Of The Erdos Harmonic Series . . . . . . . . . . . . . . . . . . . . . . 34 FELICE RUSSO, An Experimental Evidence On The Validity Of Third Smarandache Conjecture On Primes . . . . . . . . . . . . . 38 FELICE RUSSO, Some Results About Four Smarandache U-Product Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 42 FELICE RUSSO, On An Unsolved Question About The Smarandache SquarePartial-Digital Subsequences . . . . . . . . . . . . 50 CHARLES ASHBACHER, On A Conjecture By Russo . . . .. . . . . . 53 SEBASTIAN MARTIN RUIZ, A Functional Recurrence To Obtain The Prime Numbers Using The Smarandache Coprime Function . . .. . . . . 56 SEBASTIAN MARTIN RUIZ, The General Term Of The Prime Number Sequence And The Smarandache Prime Function . . . . . . . . 59 SEBASTIAN MARTIN RUIZ, Expressions Of The Smarandache Coprime Function . . . . . . . . . . . . . . . . . . . . . . . . . . 62 MLADEN V. VASSILEV-MISSANA, KRASSIMIR ATANASSOV, Note On the Diophantine Equation 2x2-3y2=p . . . . . . . . . . . . . . . 64 HRISTO ALADJOV, KRASSIMIR ATANASSOV, Remark On The 62-th Smarandache Problem . . . . . . . . . . . . . . . . . . . . 69 MAOHUA LE, The Integral Values Of log S(nk) . . . . . . . . . 71 kn RONGI CHEN, MAOHUA LE, On The Functional Equation S(n)2+S(n)=kn 73 ZHONG LI, MAOHUA LE, On The Functional Equation Z(n)+φ(n)=d(n) 77 M. BENCZE, Smarandache Relationships And Subsequences . . . . 79 SYLVESTER SMITH, A Set Of Conjectures On Smarandache Sequences 86 FANEL IACOBESCU, Smarandache Partition Type And Other Sequences 93 HELEN 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 . . . . 107 CHARLES 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 . . . . . . . . . . 117 324

CHARLES ASHBACHER, Some Problems Concerning The Smarandache Deconstructive Sequence . . . . . . . . . . . . . . . . . . 120 EMIL BURTON, S-Primality Degree Of A Number And S-Prime Numbers . . . . . . . . . . . . . . . . . . . .. . . . . . . . 123 E. RADESCU, Some Elementary Algebraic Considerations Inspired By Smarandache Type Functions (II) . . . . . . . . . . . . . . 124 RONGJI CHEN, On The Functional Equation (S(n))r + (S(n))r-1 + … + S(n) = n . . . . . . . . . . . . . . . . . . . . . . . . . 128 ZHONG LI, Solution Of Two Questions Concerning The Divisor Function And The Pseudo-Smarandache Function . . . . . . . 132 J. SANDOR, On A Conjecture Of Smarandache On Prime Numbers . 136 FLORIAN LUCA, On The Smarandache Irrationality Conjecture . . 137 J. SANDOR, A Note On S(n), Where n Is An Even Perfect Number 139 STEVEN FINCH, Moments Of The Smarandache Function . . . . . . 140 GHEORGHE DINULESCU, The Intangible Absolute Truth . . . . . .. 142 Y. V. CHEBRAKOV, Advance Of Smarandache Approach To Solving Systems Of Diophantine Equations . . . . . . . . . . . . . . 144 Y. V. CHEBRAKOV, V. V. SHMAGIN, Analytical Approach To Description Of Some Combinatorial And Number-Theoretic Computative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 156 I. 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 PseudoSmarandache And Classic Functions Of Number Theory . . . . 186 FLORIAN MUNTEANU, OCTAVIAN MUSTAFA, The Almost Presumable Maximality 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 . . . . . . . . . . . . 201 J. SANDOR, On Certain Generalizations Of The Smarandache Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 VASILE SELEACU, On The Numerical Function Smin-1 . . . . . . . 213 CHARLES ASHBACHER, On Numbers Where The Values Of The PseudoSmarandache Function Of It And The Reversal Are Identical . 215 AMARNATH MURTHY, Smarandache Reciprocal Partition Of Unity Sets And 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 The Generalized 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 . . . . . . . . . . . . . . . . . . . . . . . 275 325

AMARNATH MURTHY, Some More Ideas On Smarandache Factor Partitions . . . . . . . . . . . . . . . . . . . . . . . AMARNATH MURTHY, A Note On The Smarandache Divisor Sequences AMARNATH MURTHY, Algorithm For Listing Of Smarandache Factor Partitions . . . . . . . . . . . . . . . . . . . . . . . . AMARNATH MURTHY, Expansion Of xn In Smarandache Terms Of Permutations . . . . . . . . . . . . . . . . . . . . . . . AMARNATH MURTHY, Miscellaneous Results And Theorems On Smarandache Terms And Factor Partitions . . . . . . . . . AMARNATH MURTHY, Smarandache Maximum Reciprocal Representation Function . . . . . . . . . . . . . . . . . . . . . . . . . AMARNATH MURTHY, Open Problems And Conjectures On The Factor/Reciprocal Partition Theory . . . . . . . . . . . . AMARNATH MURTHY, Smarandache Reciprocal Function And An Elementary Inequaliy . . . . . . . . . . . . . . . . . . . AMARNATH MURTHY, Some Funny Examples Of Smarandache Lucky Methods In Algebra, Trigonometry, And Calculus . . . . . . REVIEWS: Book Reviews (CHARLES ASHBACHER, TONNY SHANNON) . . . .

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