DOCSLIB.ORG

Observations on Icosagonal Pyramidal Number

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Observations on Icosagonal Pyramidal Number International Refereed Journal of Engineering and Science (IRJES) ISSN (Online) 2319-183X, (Print) 2319-1821 Volume 2, Issue 7 (July 2013), PP. 32-37 www.irjes.com Observations On Icosagonal Pyramidal Number 1M.A.Gopalan, 2Manju Somanath, 3K.Geetha, 1Department of Mathematics, Shrimati Indira Gandhi college, Tirchirapalli- 620 002. 2Department of Mathematics, National College, Trichirapalli-620 001. 3Department of Mathematics, Cauvery College for Women, Trichirapalli-620 018. ABSTRACT:- We obtain different relations among Icosagonal Pyramidal number and other two, three and four dimensional figurate numbers. Keyword:- Polygonal number, Pyramidal number, Centered polygonal number, Centered pyramidal number, Special number MSC classification code: 11D99 I. INTRODUCTION Fascinated by beautiful and intriguing number patterns, famous mathematicians, share their insights and discoveries with each other and with readers. Throughout history, number and numbers [2,3,7-15] have had a tremendous influence on our culture and on our language. Polygonal numbers can be illustrated by polygonal designs on a plane. The polygonal number series can be summed to form “solid” three dimensional figurate numbers called Pyramidal numbers [1,4,5 and 6] that can be illustrated by pyramids. In this communication we 32 20 65n n n deal with Icosagonal Pyramidal numbers given by p and various interesting relations n 2 among these numbers are exhibited by means of theorems involving the relations. Notation m pn = Pyramidal number of rank n with sides m tmn, = Polygonal number of rank n with sides m jaln = Jacobsthal Lucas number ctmn, = Centered Polygonal number of rank n with sides m m cpn = Centered Pyramidal number of rank n with sides m gn = Gnomonic number of rank n with sides m pn = Pronic number carln = Carol number mern = Mersenne number, where n is prime culn = Cullen number Than = Thabit ibn kurrah number II. INTERESTING RELATIONS 20 6 5 1) pn32 p n p n t3, n Proof 20 3 2 3 2 2 2pn 4 n 3 n n 2 n n 4 n n 20 5 6pn 2 p n 4 t3, n www.irjes.com 32 | Page Observations on Icosagonal Pyramidal Number 20 5 5 2) pn n cp n p n Proof 20 3 3 2 2pn 5 n 3 n n n 2 n 55 2cpnn 2 p 2 n 20 5 5 pn n cp n p n 20 20 3) pn11 p n 21 g n Proof 20 3 2 pn1 6 n 19 n 15 n 2 20 3 2 pn1 6 n 19 n 7 n 5 20 20 2pnn11 2 p 8 n 2 20 20 pnn11 p 2 2 n 1 1 4) The following triples are in arithmetic progression 8 14 20 i) pn,, p n p n Proof 3 2 3 2 20 8 6n n 5 n 2 n n n pp nn 22 4n32 2 n 6 n 2 2 14 2 pn 4 12 20 ii) pn,, p n p n Proof 3 2 3 2 4 20 2n 3 n n 6 n n 5 n pp nn 62 10n32 3 n 14 n 2 6 12 2 pn 12 16 20 iii) pppnnn,, Proof 3 2 3 2 12 20 10n 3 n 7 n 6 n n 5 n pp nn 62 14n32 3 n 11 n 2 6 16 2 pn www.irjes.com 33 | Page Observations on Icosagonal Pyramidal Number 20 5 5) pn50 p n t3, n Proof 20 5 3 2 2 2pnn 2 p 4 n n 5 n n 5 4 2ptnn 5 2 3, N 20 pn 4 6) 2 5 6 ptNN 3, n1 n Proof N 20 p 2 2n 6nn 5 n1 n 2 65nn 20nn 1 7) 2p 2 Tha2nn 1 2 mer 2n Proof 20 3n 2 n n 2p 6 2 2 5 2 2n 2p20 n 2nn 2 2 3 2 1 2 1 2 2n 21Tha2nn mer 8) The following equation represents Nasty number 20 15 6 i) 12 pn cp n 2 p n n Proof 32 20 65n n n p n 2 53n3 n n 3 n 2 n 22 2 15 6 n cp 2 p n nn2 2 20 15 6 n p cp 2 p n n n n 2 Multiply by 12, then the equation represents Nasty number 2 p20 ii) n n 5 n Proof www.irjes.com 34 | Page Observations on Icosagonal Pyramidal Number 20 3 2 2pn 6 n n 5 n 20 2 p 2 n nn 56 n p20 9) n n s 24 g n nn Proof 20 p 2 n 65nn n 2 6n 6 n 1 2 2 n 1 n 4 p20 n n s 24 g n nn 20 10) 2pn 9 n 2 nct12, n t 7, n Proof 2 20 6n 6 n 1 5 n 6 pn n 22 5n2 3 n 9 n nct 12,n 22 9n nct t 12,nn 7, 2 20 2pn 9 n 2 nct12, n t 7, n 20 11 11) 2pn2 4 p n t 66, n 42 mod107 Proof 20 3 2 2pn2 6 n 37 n 69 n 42 3 2 2 2 3n n 2 n 35 n 34 n 107 n 42 11 4pnn t66, 107 n 42 20 11 2pn2 4 p n t 66, n 42 mod107 20 20 12) pn3 p n 3 2 t 56, n 29 g n 176 Proof 20 3 2 2pn3 6 n 55 n 163 n 156 20 3 2 2pn3 6 n 53 n 151 n 138 20 20 2 2 pnn33 p 2 54 n 6 n 147 20 20 2 pnn33 p 54 n 52 n 29 2 n 1 176 www.irjes.com 35 | Page Observations on Icosagonal Pyramidal Number N 20 13) 2 pn t3, N 6 t 3, N 5 sq N n1 Proof N 20 3 2 2pn 6 n n 5 n n1 65t3,NNN sq t 3, 20 8 12 18 9 14) 4pn 7 n 6 cp n cp n cp n 2 cp n 4 t3, n Proof 20 8 12 2pn 6 cp n cp n 2 t3, n 4 n (1) 20 18 9 2pn cp n 2 cp n 2 t3, n 3 n (2) Add equation (1) and (2), we get 2p20 2n 15) jal2n 2 Tha 2 n carl8 n mer n 2n Proof 2p20 n 2nn 2 6 2 2 5 2n 2n 2 n 2 n n 1 n 42 132 12 2 1218 20 14 16) pnn p n is a cubic integer Proof 3 2 3 2 20 14 6n n 5 n 4 n n 3 n pp nn 22 3 nn 20 14 3 pnn p n n 20 6 17) 24PENn p n 66 p n 7 n p n is biquadratic integer Proof 20 4 3 2 24PENnn p n 66 n n 6 n 4 3 2 n 66 n ( n n ) 7 n 20 6 4 24PENn p n 66 p n 7 n p n n www.irjes.com 36 | Page Observations on Icosagonal Pyramidal Number REFERENCES [1]. Beiler A. H., Ch.18 in Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, New York, Dover, (1996), 184-199. [2]. Bert Miller, Nasty Numbers, The Mathematics Teachers, 73(9), (1980), 649. [3]. Bhatia B.L., and Mohanty Supriya, (1985) Nasty Numbers and their characterization, Mathematical Education, I (1), 34-37. [4]. Bhanu Murthy T.S., (1990)Ancient Indian Mathematics, New Age International Publishers Limited, New Delhi. [5]. Conway J.H., and Guy R.K., (1996) The Book of Numbers, New York Springer – Verlag, 44-48. [6]. Dickson L.E., (1952), History of the Numbers, Chelsea Publishing Company, New York. [7]. Gopalan M.A., and Devibala S., (2007), “On Lanekal Number”, Antarctica J. Math, 4(1), 35-39. [8]. Gopalan M.A., Manju Somanath, and Vanitha N., (2007), “A Remarkable Lanekel Sequence”, Proc. Nat. Acad. Sci. India 77(A), II,139-142. [9]. Gopalan M.A., Manju Somanath, and Vanitha N., (2007),“On R2 Numbers”, Acta Ciencia Indica, XXXIIIM(2), 617-619. [10]. Gopalan M.A., and Gnanam A., (2008), “Star Numbers”, Math. Sci. Res.J., 12(12), 303-308. [11]. Gopalan M.A., and Gnanam A.,(2009),“Magna Numbers”, Indian Journal of Mathematical Sciences, 5(1), 33-34. [12]. Gopalan M.A., and Gnanam A., (2008) “A Notable Integer Sequence”, Math. Sci.Res. J., 1(1) 7-15. [13]. Horadam A.F., (1996), Jacobsthal Representation Numbers, Fib. Quart., 34, 40-54. [14]. Meyyappan M., (1996), Ramanujan Numbers, S.Chand and Company Ltd., First Edition. [15]. Shailesh Shirali, (1997), Primer on Number Sequences, Mathematics Student, 45(2), 63-73. www.irjes.com 37 | Page .
Load more

Recommended publications
  • Triangular Numbers /, 3,6, 10, 15, ", Tn,'" »*"
    TRIANGULAR NUMBERS V.E. HOGGATT, JR., and IVIARJORIE BICKWELL San Jose State University, San Jose, California 9111112 1. INTRODUCTION To Fibonacci is attributed the arithmetic triangle of odd numbers, in which the nth row has n entries, the cen- ter element is n* for even /?, and the row sum is n3. (See Stanley Bezuszka [11].) FIBONACCI'S TRIANGLE SUMS / 1 =:1 3 3 5 8 = 2s 7 9 11 27 = 33 13 15 17 19 64 = 4$ 21 23 25 27 29 125 = 5s We wish to derive some results here concerning the triangular numbers /, 3,6, 10, 15, ", Tn,'" »*". If one o b - serves how they are defined geometrically, 1 3 6 10 • - one easily sees that (1.1) Tn - 1+2+3 + .- +n = n(n±M and (1.2) • Tn+1 = Tn+(n+1) . By noticing that two adjacent arrays form a square, such as 3 + 6 = 9 '.'.?. we are led to 2 (1.3) n = Tn + Tn„7 , which can be verified using (1.1). This also provides an identity for triangular numbers in terms of subscripts which are also triangular numbers, T =T + T (1-4) n Tn Tn-1 • Since every odd number is the difference of two consecutive squares, it is informative to rewrite Fibonacci's tri- angle of odd numbers: 221 222 TRIANGULAR NUMBERS [OCT. FIBONACCI'S TRIANGLE SUMS f^-O2) Tf-T* (2* -I2) (32-22) Ti-Tf (42-32) (52-42) (62-52) Ti-Tl•2 (72-62) (82-72) (9*-82) (Kp-92) Tl-Tl Upon comparing with the first array, it would appear that the difference of the squares of two consecutive tri- angular numbers is a perfect cube.
    [Show full text]
  • On Cullen Numbers Which Are Both Riesel and Sierpiński Numbers
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Number Theory 132 (2012) 2836–2841 Contents lists available at SciVerse ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt On Cullen numbers which are both Riesel and Sierpinski´ numbers ∗ Pedro Berrizbeitia a, J.G. Fernandes a, ,MarcosJ.Gonzáleza,FlorianLucab, V. Janitzio Mejía Huguet c a Departamento de Matemáticas, Universidad Simón Bolívar, Caracas 1080-A, Venezuela b Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, Mexico c Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana-Azcapotzalco, Av. San Pablo #180, Col. Reynosa Tamaulipas, Azcapotzalco, 02200, México DF, Mexico article info abstract Article history: We describe an algorithm to determine whether or not a given Received 27 January 2012 system of congruences is satisfied by Cullen numbers. We use Revised 21 May 2012 this algorithm to prove that there are infinitely many Cullen Accepted 21 May 2012 numbers which are both Riesel and Sierpinski.´ (Such numbers Available online 3 August 2012 should be discarded if you are searching prime numbers with Communicated by David Goss Proth’s theorem.) Keywords: © 2012 Elsevier Inc. All rights reserved. Cullen numbers Covering congruences Sierpinski´ numbers Riesel numbers 1. Introduction n The numbers Cn = n2 + 1 where introduced, in 1905, by the Reverend J. Cullen [1] as a particular n n case of the Proth family k2 + 1, where k = n. The analogous numbers Wn = n2 − 1 where intro- duced as far as we know, in 1917, by Cunningham and Woodall [3].
    [Show full text]
  • Input for Carnival of Math: Number 115, October 2014
    Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144.
    [Show full text]
  • Newsletter 91
    Newsletter 9 1: December 2010 Introduction This is the final nzmaths newsletter for 2010. It is also the 91 st we have produced for the website. You can have a look at some of the old newsletters on this page: http://nzmaths.co.nz/newsletter As you are no doubt aware, 91 is a very interesting and important number. A quick search on Wikipedia (http://en.wikipedia.org/wiki/91_%28number%29) will very quickly tell you that 91 is: • The atomic number of protactinium, an actinide. • The code for international direct dial phone calls to India • In cents of a U.S. dollar, the amount of money one has if one has one each of the coins of denominations less than a dollar (penny, nickel, dime, quarter and half dollar) • The ISBN Group Identifier for books published in Sweden. In more mathematically related trivia, 91 is: • the twenty-seventh distinct semiprime. • a triangular number and a hexagonal number, one of the few such numbers to also be a centered hexagonal number, and it is also a centered nonagonal number and a centered cube number. It is a square pyramidal number, being the sum of the squares of the first six integers. • the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): 91 = 6 3+(-5) 3 = 43+33. • the smallest positive integer expressible as a sum of six distinct squares: 91 = 1 2+2 2+3 2+4 2+5 2+6 2.
    [Show full text]
  • Second Grade Unit 1: Extending Base Ten Understanding
    Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Second Grade Unit 1: Extending Base Ten Understanding These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. Georgia Department of Education Georgia Standards of Excellence Framework GSE Extending Base Ten Understanding Unit 1 Unit 1: Extending Base Ten Understanding TABLE OF CONTENTS Overview……………………………………………………………………………..... 3 Standards for Mathematical Practice …………………………………………………. 5 Standards for Mathematical Content …………………………………………………. 6 Big Ideas ………………………………………………………………………………. 6 Essential Questions ……………………………………………………………………….. 7 Concepts and Skills to Maintain ………………………………………………………. 7 Strategies for Teaching and Learning ……………………………………………. 8 Selected Terms and Symbols ………………………………………………..………… 10 Task Types ………………………………………………………………...………….. 12 Task Descriptions …………………………………………………………………….. 13 Intervention Table………………….………………………………………………….. 15 Straws! Straws! Straws!.........................................................................… 16 Where Am I on the Number Line? …………………………………… 23 I Spy a Number ………………………………………………………. 31 Number Hop …………………………………………………………… 38 Place Value Play …………………………………………………….... 44 The Importance of Zero ………………………………………………. 51 Base Ten Pictures …………………………………………………….. 62 Building Base Ten Numbers …………………………………………. 69 What's My Number? …………………………………………………. 74 Capture the Caterpillar ……………………………………………….. 79 Formative Assessment Lesson ……………………………………….. 88 Fill the Bucket ………………………………………………………..
    [Show full text]
  • Cullen Numbers with the Lehmer Property
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0 CULLEN NUMBERS WITH THE LEHMER PROPERTY JOSE´ MAR´IA GRAU RIBAS AND FLORIAN LUCA Abstract. Here, we show that there is no positive integer n such that n the nth Cullen number Cn = n2 + 1 has the property that it is com- posite but φ(Cn) | Cn − 1. 1. Introduction n A Cullen number is a number of the form Cn = n2 + 1 for some n ≥ 1. They attracted attention of researchers since it seems that it is hard to find primes of this form. Indeed, Hooley [8] showed that for most n the number Cn is composite. For more about testing Cn for primality, see [3] and [6]. For an integer a > 1, a pseudoprime to base a is a compositive positive integer m such that am ≡ a (mod m). Pseudoprime Cullen numbers have also been studied. For example, in [12] it is shown that for most n, Cn is not a base a-pseudoprime. Some computer searchers up to several millions did not turn up any pseudo-prime Cn to any base. Thus, it would seem that Cullen numbers which are pseudoprimes are very scarce. A Carmichael number is a positive integer m which is a base a pseudoprime for any a. A composite integer m is called a Lehmer number if φ(m) | m − 1, where φ(m) is the Euler function of m. Lehmer numbers are Carmichael numbers; hence, pseudoprimes in every base. No Lehmer number is known, although it is known that there are no Lehmer numbers in certain sequences, such as the Fibonacci sequence (see [9]), or the sequence of repunits in base g for any g ∈ [2, 1000] (see [4]).
    [Show full text]
  • On Ternary Cubic Equation
    ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 9, March 2014 ON TERNARY CUBIC EQUATION P. Thirunavukarasu, S. Sriram Assistant Professor -P.G & Research Department of Mathematics, Periyar E.V.R College Tiruchirappalli – 620 023, Tamilnadu, India Assistant Professor–P.G & Research Department of Mathematics, National College, Tiruchirappalli – 620 001, Tamilnadu, India equations are analyzed for the non-trivial integral Abstract we obtain the non-trivial integral solutions for the solutions. These results have motivated us to search for ternary cubic equation non-trivial integral solutions of their varieties of ternary cubic Diophantine equation. This paper concerns with the problem of determining non-trivial integral solutions of . the equation with three unknowns given by A few interesting relations among the solutions are presented. Index Terms: Ternary Cubic, integral solutions, Pell’s explicit integral solutions of the above equation are form, nasty numbers presented. A few interesting relations among the solutions Notations are obtained. Oblong number of rank n obln n n 1 II. METHOD OF ANALYSIS Tetrahedral number of rank The ternary cubic equation under consideration is n n12 n n Tet n 6 Triangular number of rank (1) Taking (2) Polygonal number of rank n with sides nm12 (3) m tnmn, 1 2 We get Square pyramidal number of rank n n1 2 n 1 (4) Again taking the transformation n Sqpn 6 Pentagonal pyramidal number of rank Star number = and apply in (4) we get (5) 2 Stella Octangula number = St. oct n 2 n 1 n It is well known that the general form of the integral solutions of the Pellian equation.
    [Show full text]
  • ~Umbers the BOO K O F Umbers
    TH E BOOK OF ~umbers THE BOO K o F umbers John H. Conway • Richard K. Guy c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a re­ trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus, an imprint of Springer-Verlag New York, Inc. Copernicus Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY lOOlO Library of Congress Cataloging in Publication Data Conway, John Horton. The book of numbers / John Horton Conway, Richard K. Guy. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8488-8 e-ISBN-13: 978-1-4612-4072-3 DOl: 10.l007/978-1-4612-4072-3 1. Number theory-Popular works. I. Guy, Richard K. II. Title. QA241.C6897 1995 512'.7-dc20 95-32588 Manufactured in the United States of America. Printed on acid-free paper. 9 8 765 4 Preface he Book ofNumbers seems an obvious choice for our title, since T its undoubted success can be followed by Deuteronomy,Joshua, and so on; indeed the only risk is that there may be a demand for the earlier books in the series. More seriously, our aim is to bring to the inquisitive reader without particular mathematical background an ex­ planation of the multitudinous ways in which the word "number" is used.
    [Show full text]
  • Fibonacci S-Cullen and S-Woodall Numbers
    1 2 Journal of Integer Sequences, Vol. 18 (2015), 3 Article 15.1.4 47 6 23 11 Fibonacci s-Cullen and s-Woodall Numbers Diego Marques Departamento de Matem´atica Universidade de Bras´ılia Bras´ılia, Brazil [email protected] Ana Paula Chaves Instituto de Matem´atica e Estat´ıstica Universidade Federal de Goi´as Goi´as, Brazil [email protected] Abstract m The m-th Cullen number Cm is a number of the form m2 + 1 and the m-th m Woodall number Wm has the form m2 1. In 2003, Luca and St˘anic˘aproved that − the largest Fibonacci number in the Cullen sequence is F4 = 3 and that F1 = F2 = 1 are the largest Fibonacci numbers in the Woodall sequence. Very recently, the second m author proved that, for any given s> 1, the equation Fn = ms 1 has only finitely many solutions, and they are effectively computable. In this note,± we shall provide the explicit form of the possible solutions. 1 Introduction m A Cullen number is a number of the form m2 +1 (denoted by Cm), where m is a nonnegative integer. This sequence was introduced in 1905 by Father J. Cullen [2] and it was mentioned in the well-known book of Guy [5, Section B20]. These numbers gained great interest in 1976, when Hooley [7] showed that almost all Cullen numbers are composite. However, despite being very scarce, it is still conjectured that there are infinitely many Cullen primes. 1 In a similar way, a Woodall number (also called Cullen number of the second kind) is m a positive integer of the form m2 1 (denoted by Wm).
    [Show full text]
  • Elementary Proof That an Infinite Number of Cullen Primes Exist
    Elementary Proof that an Infinite Number of Cullen Primes Exist Stephen Marshall 21 February 2017 Abstract This paper presents a complete proof of the Cullen Primes are infinite, even though only 16 of them have been found as of 21 Feb 2017. We use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer m: m = (p-1)!( + ) + + We use this proof for d = to prove the infinitude of Cullen prime numbers. The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Cullen Prime Conjecture possible. Introduction James Cullen, (19 April 1867 – 7 December 1933) was born at Drogheda, County Louth, Ireland. At first he was educated privately, then he studied pure and applied mathematics at the Trinity College, Dublin for a while, he went to 1 Mungret College, Limerick, before deciding to become a Jesuit, studying in England in Mansera House, and St. Mary's, and was ordained as a priest on 31 July 1901. In 1905, he taught mathematics at Mount St. Mary's College in Derbyshire and published his finding of what is now known as Cullen numbers in number theory. In number theory of prime numbers, a Cullen number is a natural number of the form n Cn = n2 + 1 Cullen numbers were first studied by James Cullen in 1905.
    [Show full text]
  • Collection of Problems on Smarandache Notions", by Charles Ashbacher
    Y ---~ y (p + 1,5(p + 1)) p p + 1 = Ex-b:u.s "1:T::n.i."'I7ex-lI!d.~ Pre._ •Va.:U. ~996 Collect;io:n. of Problern..s O:n. Srn..a.ra:n.dache N"o"i;io:n.s Charles Ashbacher Decisio:n.rn..a.rk 200 2:n.d A:v-e. SE Cedar Rapids, IA. 52401 U"SA Erhu.s U":n.i"V"ersity Press Vall 1996 © Cha.rles Ashbacher & Erhu.s U":n.i"V"ersity Press The graph on the first cover belongs to: M. Popescu, P. Popescu, V. Seleacu, "About the behaviour of some new functions in the number theory" (to appear this year) . "Collection of Problems on Smarandache Notions", by Charles Ashbacher Copyright 1996 by Charles Ashbacher and Erhus University Press ISBN 1-879585-50-2 Standard Address Number 297-5092 Printed in the United States of America Preface The previous volume in this series, An Introduction to the Smarandache Function, also by Erhus Cniversity Press, dealt almost exclusively with some "basic" consequences of the Smarandache function. In this one, the universe of discourse has been expanded to include a great many other things A Smarandache notion is an element of an ill-defined set, sometimes being almost an accident oflabeling. However, that takes nothing away from the interest and excitement that can be generated by exploring the consequences of such a problem It is a well-known cliche among writers that the best novels are those where the author does not know what is going to happen until that point in the story is actually reached.
    [Show full text]
  • Arxiv:2010.10014V1 [Math.NT] 20 Oct 2020 As Suethat Assume Where Where Nqeydtrie.W Altesqec ( Sequence the Call We Determined
    CULLEN NUMBERS IN SUMS OF TERMS OF RECURRENCE SEQUENCE N.K. MEHER AND S. S. ROUT Abstract. Let (Un)n≥0 be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer ℓ, let ℓ 2ℓ + 1 be a Cullen number. Recently in · [3], generalized Cullen numbers in terms of linear recurrence sequence (Un)n≥0 under certain weak assumptions has been studied. However, there is an error in their proof. In this paper, we generalize their work, as well as our result fixes their error. In particular, for a given polynomial Q(x) Z[x] we consider the Diophantine equation ℓ ∈ Un1 + + Unk = ℓ x + Q(x), and prove effective finiteness result. Furthermore, we demonstrate··· our method· by an example. 1. Introduction Let r be a positive integer. The linear recurrence sequence (Un)n≥0 of order r is defined as U = a U + + a U (1.1) n 1 n−1 ··· r n−r where a ,...,a Z with a = 0 and U ,...,U are integers not all zero. 1 r ∈ r 6 0 r−1 The characteristic polynomial of Un is defined by t f(x) := xr a xr−1 a = (x α )mi Z[X] (1.2) − 1 −···− r − i ∈ i=1 Y arXiv:2010.10014v1 [math.NT] 20 Oct 2020 where α1,...,αt are distinct algebraic numbers and m1,...,mt are positive integers. Then Un (see e.g. Theorem C1 in part C of [14]) has a nice representation of the form t U = f (n)αn for all n 0, (1.3) n i i ≥ i=1 X where f (x) is a polynomial of degree m 1 (i = 1,...,t) and this representation is i i − uniquely determined.
    [Show full text]