Smarandache Notions Journal, Vol. 14

Vol. 14, 2004

ISSN 1084-2810

SMARANDACHE NOTIONS JOURNAL

UNIVERSITY COLLEGE CORK

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), "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. M. Khoshnevisan, School of Accounting and Finance, Griffith University, Australia. Maohua Le, Department of Mathematics, Zhanjiang Normal College, 29 Cunjin Road, Chikan, Zhanjiang, Guangdong, P. R. China. Chris Nash, 5000 Bryan Station Road #5, Lexington, KY 40516-9505, USA. Sebastian Martin Ruiz, Avda. De Regla, 43, Chipiona 11550, Cadiz, Spain.

Copyright 2004 by Sabin & Tatiana Tabirca, University College Cork, Ireland.

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ASSOCIATE EDITORIAL BOARD: Azmy Ariff, PO Box 444, Jalan Sultan, Petaling Jaya 46750, Malaysia. Charles Ashbacher, Charles Ashbacher Technologies, Box 294, 119 Northwood Drive, Hiawatha, Iowa 52233, USA. Krassimir Atanassov, Bulgarian Academy of Sciences, PO Box 12 MRL, Sofia, Bulgaria. Ion Balacenoiu, Department of Mathematics, University of Craiova, Romania. Mihaly Bencze, Str. H|rmanului 6, 2212 S|cele 3, Jud. BraÕov, Romania. Henry Bottomley, 5 Leydon Close, London SE16 5PF, U.K. Micha Fleuren, Acaciastraat 16, 6598 BC Heyen, The Netherlands. H. Gunarto, Ritsumeikan Asia-Pacific University, 1-1 Jumonjibaru, Beppu-shi 874-8577, Oita-ken, Japan. Henry Ibstedt, Glimminge 2036, 280 60 Broby, Sweden. Howard Iseri, Department of Mathematics, Mansfield University, Mansfield, PA 16933, USA. Muneer Jebreel, SS-Math-Hebron, UNRWA, Field Education Officer, Box 19149, Jerusalem, Israel. W. B. Vasantha Kandasamy, Indian Institute of Technology, Madras, Chennay 600 036, India. 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. 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. Delfim F. M. Torres, Department of Mathematics, University of Aveiro, 3810-193, Portugal. Angela Vasiu, Mathematics Department, Babes-Bolyai University, Cluj-Napoca, Romania. Bouzza El Wahbi, Départment de Mathématques et Informatiques, Faculté des Sciences, B.P. 2121 Tétouan, Morocco. Zhang Wenpeng, Department of Mathematics, Xi’an University of Technology, Xi’an, Shaanxi, P.R. China. Lamarr Widmer, Department of Mathematics, Messiah College, Grantham, PA 17027, USA. INDICATION TO AUTHORS

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Authors of papers concerning any of Smarandache type functions, numbers, sequences, integer algorithms, paradoxes, Non-Euclidean geometries, etc. are encouraged to submit manuscripts to Editors: Prof. Sabin & Tatiana Tabirca Department of Mathematics University College Cork, Ireland. 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 XP (or lower), WordPerfect 7.0 (or lower), PDF 6.0 or lower.

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CONTENTS

MATHEMATICS: PAULO D. F. GOUVEIA, DELFOM F. M. TORRES, Smarandache Sequences: Explorations and Discoveries with a Computer Algebra System …...…………………………. 5 HENRY IBSTEDT, Palindrome Studies (Part I: the Palindrome Concept and Its Applications to Prime Numbers) ………………………………………………………….....…23 A. S. MUKTIBODH, V. M. WAGH, Chains of Smarandache Semifields …………………...36 CHARLES ASHBACHER, On Numbers That Are Pseudo-Smarandache and Smarandache Perfect ……………………………………………………………………………………….…..40 A. W. VYAWAHARE, K. M. PUROHIT, Near Pseudo-Smarandache Function …………....42 ZHANG TIANPING, On the k-Power Free Number Sequence ………………………………62 ZHU WEIYI, On the k-Power Complement and k-Power Free Number Sequence …………66 HE XIAOLIN, GUO JINBAO, On the 80-th Problem of F. Smarandache (I) ………………70 HE XIAOLIN, GUO JINBAO, On the 80-th Problem of F. Smarandache (II) …………..…74 MUNEER JEBREEL KARAMA, Smarandache Concatenated Magic Squares ……………..80 ZHENG JIANFENG, Proof of Functional Smarandache Iterations ……….. ………………84 ZHENG JIANFENG, On the Inferior and Superior k-th Power Part of a Positive Integer and Divisor Function ………………………………………………………………………………..88 A. A. MAJUMDAR, H. GUNARTO, On Some Recurrence Type Smarandache sequences …92 ADRIAN VASIU, ANGELA VASIU, The Fulfilled Euclidean Plane ………………………113 DELFIM F. M. TORRES, Números Felizes e Sucessōes de Smarandache Digressōes com o Maple ………………………………………………………………………………………….119 JOHN PERRY, Calculating the Smarandache Numbers ……………………………………124 JÓZSEF SÁNDOR, On Additive Analogues of Certain Arithmetic Functions ……………..128 SABIN TABIRCA, KIERAN REYNOLDS, Recursive Prime Numbers ……………………..133 MUNEER JEBREEL, Smarandache Sequence of Happy Cube Numbers ………………….139 ZHU WEIYI, On the Divisor Product Sequence ……………………………………………..144 ZHANG TIANPING, On the Cubic Residues Numbers and k-Power Complement Numbers ……………………………………………………………………………………………….....147 MAOHUA LE, A Conjecture Concerning the Smarandache Dual Function ………………153 LIU HONGYAN, LOU YUANBING, A Note on the 29-th Smarandache’s Problem ……....156 XU ZHEFENG, On the k-Full Number Sequence …………………………………………..159 LIU HONGYAN, ZHANG WENPENG, A Note on the 57-th Smarandache’s Problem …...164 SEBASTIAN MARTIN RUIZ, Diverse Algorithms to Obtain Prime Numbers Based on the Prime Function of Smarandache …………………………………………………………….166 414

LIU HONGYAN, ZHANG WENPENG, On the Simple Numbers and the Mean Value Properties ……………………………………………………………………………………...171 MOHUA LE, The Divisibility of the Smarandache Combinatorial Sequence of Degree Two ………………………………………………………………………………………………….176 MAOHUA LE, The Smarandache φ–Sequence …………..………………………………….178 MAOHUA LE, Two Functional Equations …………………………………………………..180 MAOHUA LE, Two Formulas for Smarandache LCM Ratio Sequences …………………..183 MAOHUA LE, An Equation Concerning the Smarandache LCM Functions ……………...186 GAO JING, LIU HUANING, On the 45-th Smarandache’s Problem ………………………189 GAO JING, LIU HUANING, On the Mean Value of Smarandache Double Factorial Function ……………………………………………………………………………………….193 CHARLES ASHBACHER, LORI NEIRYNCK, The Density of Generalized Smarandache Palindromes ……………………………………………………………………………………196 ZHU WEIYI, On the Cube Free Number Sequence …………………………………………199 GAO JING, On the 49-th Smarandache’s Problem ………………………………………….203 HOWARD ISERI, Paper Models of Surfaces with Curvature Creative Visualization Labs ..205 CHARLES T. LE, BERND HUTSCHENREUTHER, Die Smarandache’sche Klasse von Paradoxien ………………………………………………………………………………. …...211 BOUAZZA EL WAHBI, Convergence of the Smarandache General Continued Fraction ...213 MUNEER JEBREEL, The first 10-th Smarandache Symmetric Numbers …………………215 LIU HUANING, ZHANG WENPENG, On the 57-th Smarandache’s Problem ……………218 MAOHUA LE, Some Problems Concerning the Smarandache Square Complementary Function (I) ……………………………………………………………………………………220 JASON EARLS, Some Results Concerning the Smarandache Deconstructive Sequence ….222 LOU YUANBING, An Asymptotic Formula Involving Square Complement Numbers …….227 FELICE RUSSO, On A Problem Concerning the Smarandache Left-Right Sequences …...230 CHARLES ASHBACHER, Some Properties of the Happy Numbers and the Smarandache HSequence ……………………………………………………………………………………….236 JASON EARLS, Some Smarandache-Type Sequences and Problems Concerning Abundant and Deficient Numbers ……………………………………………………………………..…243 JASON EARLS, On Smarandache Repunit n Numbers …………………………………….251 CHARLES ASHBACHER, Some Simple Advantages of Reasoning in Intuitionistic Neutrosophic Logic ……………………………………………………………………………257 JASON EARLS, Smarandache Iterations of the First Kind on Functions Involving Divisors and Prime Numbers …………………………………………………………………………...259 JASON EARLS, The Smarandache Sum of Composites between Factors Function ………265 YAO WEILI, On the Mean Value of m-th Power Free Numbers …………………………...271 415

JASON EARLS, A Note on the Smarandache Divisors of Divisors Sequence and Two similar Sequences ……………………………………………………………………………………...274 SEBASTIAN MARTIN RUIZ, Formula to Obtain the Next Prime Number in Any Increasing Sequence of Positive Integer Numbers ……………………………………………………….276 W. B. VASANTHA KANDASAMY, Smarandache Non-Associative (SNA-) Rings …..…….281 W. B. VASANTHA KANDASAMY, Smarandache Pseudo-Ideals …………………………..294 HOWARD ISERI, Impulse Gauss Curvature ………………………………………………..299 W. B. VASANTHA KANDASAMY, Smarandache-Galois Fields …………………………...310 W. B. VASANTHA KANDASAMY, Smarandache Near-Rings and Their Generalizations ..315 W. B. VASANTHA KANDASAMY, Smarandache Semi-Automaton and Automaton ……..320 W. B. VASANTHA KANDASAMY, Smarandache Zero Divisors …………………………...325 MAOHUA LE, Some Problems Concerning the Smarandache Square Complementary Function (II) ……………………………………………………………………………..……330 MAOHUA LE, Some Problems Concerning the Smarandache Square Complementary Function (III) …………………………….……………………………………………………333 MAOHUA LE, Some Problems Concerning the Smarandache Square Complementary Function (IV) ………………………………………….………………………………………335 MAOHUA LE, Some Problems Concerning the Smarandache Square Complementary Function (V) …………………………………………………………...………………………338 MAOHUA LE, The Smarandache Combinatorial Sequences ……………………………….341 MAOHUA LE, A Conjecture Concerning Indexes of Beauty ……………………………….343 MAOHUA LE, The Smarandache-Riemann Zeta Sequence ………………………………..346 LI HAILONG, On Third Power Mean Values Computation of Digital Sum Function in Base n ………………………………………………………………………………………………..348 HENRY BOTTOMLEY, A Lucky Derivative ………………………………………………...352 HOWARD ISERI, A Classification of S-Lines in a Closed S-Manifold …………………….354 SEBASTIAN MARTIN RUIZ, AZMY ARIFF, A Basic Characteristic of Twin Primes and Its Generalization …………………………………………………………………………………357 SEBASTIAN MARTIN RUIZ, Formula to Obtain the Next Prime in an Arithmetic Progression …………………………………………………………………………………….363 SHYAM SUNDER GUPTA, Smarandache Sequence of Triangular Numbers …………….366

PHYLOSOPHY FENG LIU, Truth and Absolute Truth in Neutrosophic Logic ……………………………..369 FENG LIU, Toward Excitation and Inhibition in Neutrosophic Logic ……………………..381 FLAVIAN VASILE, Processes of Consciousness ……………………………………………389 CHENG-GUI HUANG, A Note on Neutrosophy and Buddhism ……………………………398 416

PHYSICS: HOMER B. TILTON, Reverse Engineer the Universe! ……………………………………...399 SOCIEDAD ESPAŇOLA DE EXOBIOLOGIA, HG Wells Telecom (Pregúntale al Científico) ………………………………………………………………………………………………….404 SOCIEDAD ESPAŇOLA DE EXOBIOLOGIA, Estación Central Tiempo (Pregúntale al Científico) ………………………………..…………………………………………………….407 J. HARRI T. TIAINEN, The Schrödinger Cat Thought Experiment Land (The SCTE Land) ………………………………………………………………………………………………….409

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Papers concerning any of the Smarandache type functions, sequences, numbers, algorithms, inferior/superior f-parts, magic squares, palindromes, functional iterations, semantic paradoxes, Non-Euclidean geometries, manifolds, conjectures, open problems, algebraic structures, neutrosophy, neutrosophic logic/set/probability, hypothesis that there is no speed barrier in the universe, quantum paradoxes, etc. have been selected for this volume. Contributors are from Australia, China, England, Germany, India, Ireland, Israel, Italy, Japan, Malaysia, Morocco, Portugal, Romania, Spain, USA. Most of the papers are in English, a few of them are in Spanish, Portuguese, or German. The Editors

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