Smarandache Notions Journal, Vol. 13

Vol. 13, No. 1-2-3, Spring 2002

ISSN 1084-2810

SMARANDACHE NOTIONS JOURNAL

UNIVERSITY COLLEGE CORK

1

The graph on the first cover represents a Howard Iseri Model for a Smarandache Geometry (see p. 8). This journal is an extension of the "Smarandache Function Journal", ISBN 1053-4792, Vol. 1-6, 1990-1995. SNJ is yearly published (in the Spring or Fall) in a 300-400 pages volume, and 800-1000 copies. SNJ is a referred journal: reviewed, indexed, cited by the following journals: "Zentralblatt Für 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). Referents: Murad A. AlDamen, P. O. Box 713330, Hai Nazzal 11171, Amman, Jordan. Y. V. Chebrakov, Department of Mathematics, St.-Petersburg Technical University, Nevsky 3-11, 191186, Russia. Ion Goian, Department of Algebra, Number Theory and Logic, State University of Kishinev, Republic of Moldova. Florian Luca, Instituto de Matematicas UNAM, Campus Morelia, Apartado Postal 61-3 (Xangari), CP 58 089, Morelia, Michoacan, México. Chris Nash, 5000 Bryan Station Road #5, Lexington, KY 40516-9505, USA. Carlos B. Rivera F., J. Pardavé 517, Col Lomas del Roble, San Nicolás de los Garza, CP 66450, Nuevo León, México. Sebastian Martin Ruiz, Avda. De Regla, 43, Chipiona 11550, Cadiz, Spain. Copyright 2002 by Sabin & Tatiana Tabirca, University College Cork, Ireland. ASSOCIATE EDITORIAL BOARD: Jack Allen, School of Accounting and Finance, Griffith University, Australia. Charles Ashbacher, Charles Ashbacher Technologies, Box 294, 2

119 Northwood Drive, Hiawatha, Iowa 52233, USA. Ion Balacenoiu, Department of Mathematics, University of Craiova, Romania. Mihaly Bencze, Str. H|rmanului 6, 2212 S|cele 3, Jud. BraÕov, Romania. Dragos Constantinescu, College of Arts, Rm. Valcea, Jud. Valcea, Romania. Constantin Corduneanu, Department of Mathematics, University of Texas at Arlington, USA. Henry Ibstedt, Glimminge 2036, 280 60 Broby, Sweden. Howard Iseri, Department of Mathematics, Mansfield University, Mansfield, PA 16933, USA. Maohua Le, Department of Mathematics, Zhanjiang Normal College, 29 Cunjin Road, Chikan, Zhanjiang, Guangdong, P. R. China. Feng Liu, Dept. of Management Science and Engineering, Shaanxi Economics and Trade Institute (South Campus), South Cuihua Road, Xi'an, Shaanxi, 710061, P. R. China. A.A.K. Majumdar, Department of Mathematics, Jahangirnagar University, Savar, Dhaka 1342, Bangladesh. Leonardo Fernandes Dias da Motta, Av. Conselheiro Furtado, 1574, Apt. 501, Bairro Batista Campos, Belem, PA 66040-100, Brasil. Mike Mudge, 22 Gors Fach, Pwll-Trap, St Clears, Carmarthenshire, Dyfed SA33 4AQ, United Kingdom. E. R|descu & N. R|descu, Department of Mathematics, University of Craiova, Romania. Felice Russo, Micron Technology Italy, Avezzano (Aq), Italy. Vasile Seleacu, Department of Mathematics, University of Craiova, Romania. Y. A. Shmagin, Department of Mathematics, St.-Petersburg Technical University, Nevsky 3-11, 191186, St.-Petersburg, Russia. Vera W. de Spinadel, Centro de Matematica y Diseno MAyDI, Facultad de Arquitectura, Diseno y Urbanismo, Universidad de Buenos Aires, Argentina.

INDICATION TO AUTHORS Authors of papers concerning any of Smarandache type functions, numbers, sequences, integer algorithms, paradoxes, Non-Euclidean geometries, etc. are encouraged to submit manuscripts to the Editors: Prof. Sabin & Tatiana Tabirca Department of Mathematics University College Cork, Ireland. 3

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 writing 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 2000 (or lower), WordPerfect 7.0 (or lower), PDF 4.0, or ASCII files.

4

CONTENTS

MATHEMATICS: HOWARD ISERI, Partially Paradoxist Smarandache Geometries .…………………………. 5 CLIFFORD SINGER, Engineering a Visual Field .........…………………………………... 13 J. SÁNDOR, On Smarandache’s Podaire Theorem …………………………………………. 16 J. SÁNDOR, On a Dual of the Pseudo-Smarandache Function ……………………………. 18 TATIANA TABIRCA, SABIN TABIRCA, A New Equation for the Load Balance Scheduling Based on the Smarandache f-Inferior Part Function ……………………………………….. 24 SABIN TABIRCA, TATIANA TABIRCA, Some New Results Concerning the Smarandache Ceil Function ………………………………………………………………………………..… 30 MARK FARRIS, PATRICK MITCHELL, Bounding the Smarandache Function……..….. 37 JACK ALLEN, SUKANTO BHATTACHARYA, Critical Trigger Mechanism – a Modeling Paradigm for Cognitive Science Application in the Design of Artificial Learning systems …43 PANTELIMON STANICA, GABRIELA STANICA, There are Infinitely Many Smarandache Derivations, Integrations and Lucky Numbers ……………………………………………… 48 AMARNATH MURTHY, Program for Finding out the Number of Smarandache Distinct Reciprocal Partition of Unity of a Given Length ……………………………………………. 53 FELICE RUSSO, On a Problem Concerning the Smarandache Friendly Prime Pairs …… 56 SHYAM SUNDER GUPTA, Smarandache Sequence of Happy Numbers …………………. 59 FELICE RUSSO, On a Smarandache Problem Concerning the Prime Gap ………………. 64 DAVID POWER, SABIN TABIRCA, TATIANA TABIRCA, Java Concurrent Program for the Smarandache Function ………………………………………………………………………. 72 L. Kuciuk, M. Antholy, An Introduction to the Smarandache Geometries …………………..85 W. B. VASANTHA KANDASAMY, Smarandache Semirings and Semifields ……………… 88 SEBASTIAN MARTIN RUIZ, The Sequence of Prime Numbers …………………………. 92 HENRY IBSTEDT, On a Concatenation Problem ………………………………………….. 96 HENRY IBSTEDT, On a Deconcatenation Problem ………………………………………..107 MAOHUA LE, On Smarandache Friendly Natural Number Pairs …………………………120 KRASSIMIR T. ATANASSOV, On the 17-th Smarandache Problem …………………… 124 KRASSIMIR T. ATANASSOV, On the 46-th Smarandache Problem …………………… 126 LIU HONGYAN, ZHANG WENPENG, On the Divisor Product and Proper Divisor Product Sequences …………………………………………………………………………………… 128 HENRY BOTTOMLEY, Some Smarandache-Type Multiplicative Functions …………… 134 DAVID GORSKI, The Pseudo-Smarandache Function …………………………………… 140 ZHANG WENPENG, On the Symmetric Sequence and Some of its Properties ……………150 285

ZHANG WENPENG, On the Permutation Sequence and some of its Properties ………… 153 LIU HONGYAN, ZHANG WENPENG, A Number Theoretic Function and its Mean Value Property ………………………………………………………………………………………..155 FELICE RUSSO, An Introduction to the Smarandache Square Complementary Function 160 ZHANG WENPENG, LIU DUANSEN, On the Primitive Number of Power P and its Asymptotic Property ………………………………………………………………………… 173 ZHANG WENPENG, On an Equation of Smarandache and its Integer Solutions ………. 176 ZHANG WENPENG, A New Sequence Related to Smarandache Sequences and its Mean Value Formula ……………………………………………………………………………….. 179 FELICE RUSSO, Five Properties of the Smarandache Double Factorial Function ………183 ZHANG WENPENG, YI YUAN, On a Problem of F. Smarandache ……………………… 186 ZHANG WENPENG, On a Smarandache-Lucas Base and Related Counting Function … 191 J. SÁNDOR, On a Generalized Bisector Problem ……………………………………………197 WANG YANG, ZHANG HONG LI, On a Conjecture of F. Smarandache …………………199 FENG LIU, Paradoxes Review ……………………………………………………………… 200 XIGENG CHEN, The Equation "2(k+2, S(n))= "2(k+1, S(n))+ "2(k, S(n)) …… ……..….. 204 XIGENG CHEN, The Equation $2(k+2, S(n))= $2(k+1, S(n))+ $2(k, S(n)) ……. .……….. 206 MAOHUA LE, On the Smarandache Double Factorial Function ………………………… 209 MAOHUA LE, On the Pseudo-Smarandache Square Free Function ………………………229 MAOHUA LE, The Equation S(1x2)+ S(2x3)+…+ S(n(n+1))=S(n(n+1)(n+2)/3)) ……….. 237 MAOHUA LE, Some Conjectures on Primes (I) ………………………………………….. 239 MAOHUA LE, Some Conjectures on Primes (II) ………………………………………….. 241 MAOHUA LE, Some Conjectures on Primes (III) ….…………………………………….. 243 W. B. VASANTHA KANDASAMY, Smarandache Cosets ………………………………….. 245 W. B. VASANTHA KANDASAMY, Smarandache Loops ………………………………….. 252 S. M. TABIRCA, I. PITT, D. MURPHY, A Discrete Model for Histogram Shaping ………259 JOZSEF SANDOR, On an Additive Analogue of the Function S …………………………. 266

PHYSICS: LEONARDO F. D. DA MOTTA, Hipótese de Smarandache: Evidências, Implicações Aplicações [Portuguese] ………………………………………………................................ 271 JASON WRIGHT, Superluminals and the Speed of Light .................................................... 276 FELICE RUSSO, Faster Than Light? ……………………………………………………… 281 REVIEW ……………………………………………………………………………..………. 285

286