Special Pythagorean Triangle in Relation with Pronic Numbers
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Guidance Via Number 40: a Unique Stance M
Sci.Int.(Lahore),31(4),603-605, 2019 ISSN 1013-5316;CODEN: SINTE 8 603 GUIDANCE VIA NUMBER 40: A UNIQUE STANCE M. AZRAM 7-William St. Wattle Grove, WA 6107, USA. E-mail: [email protected] ABSTRACT: The number Forty (40) is a mysterious number that has a great significance in Mathematics, Science, astronomy, sports, spiritual traditions and many cultures. Historically, many things took place labelled with number 40. It has also been repeated many times in Islam. There must be some significantly importance associated with this magic number 40. Only Allah knows the complete answer. Since Muslims are enjoined to establish good governance through a just sociomoral order (or a state) wherein they could organise their individual and collective life in accordance with the teachings of the Qur’an and the Sunnah of the Prophet (SAW). I have gathered together some information to find some guideline via significant number 40 to reform our community by eradicating corruption, exploitation, injustice and evil etc. INTRODUCTION Muslims are enjoined to establish good governance through a just sociomoral order (or a state) wherein they could organise their individual and collective life in accordance with the teachings of the Qur’an and the Sunnah of the pentagonal pyramidal number[3] and semiperfect[4] .It is the expectation of the Qur’an that its number which have fascinated mathematics .(ﷺ) Prophet 3. 40 is a repdigit in base 3 (1111, i.e. 30 + 31 + 32 + 33) adherents would either reform the earth (by eradicating , Harshad number in base10 [5] and Stormer number[6]. corruption, exploitation, injustice and evil) or lay their lives 4. -
Monthly Science and Maths Magazine 01
1 GYAN BHARATI SCHOOL QUEST….. Monthly Science and Mathematics magazine Edition: DECEMBER,2019 COMPILED BY DR. KIRAN VARSHA AND MR. SUDHIR SAXENA 2 IDENTIFY THE SCIENTIST She was an English chemist and X-ray crystallographer who made contributions to the understanding of the molecular structures of DNA , RNA, viruses, coal, and graphite. She was never nominated for a Nobel Prize. Her work was a crucial part in the discovery of DNA, for which Francis Crick, James Watson, and Maurice Wilkins were awarded a Nobel Prize in 1962. She died in 1958, and during her lifetime the DNA structure was not considered as fully proven. It took Wilkins and his colleagues about seven years to collect enough data to prove and refine the proposed DNA structure. RIDDLE TIME You measure my life in hours and I serve you by expiring. I’m quick when I’m thin and slow when I’m fat. The wind is my enemy. Hard riddles want to trip you up, and this one works by hitting you with details from every angle. The big hint comes at the end with the wind. What does wind threaten most? I have cities, but no houses. I have mountains, but no trees. I have water, but no fish. What am I? This riddle aims to confuse you and get you to focus on the things that are missing: the houses, trees, and fish. 3 WHY ARE AEROPLANES USUALLY WHITE? The Aeroplanes might be having different logos and decorations. But the colour of the aeroplane is usually white.Painting the aeroplane white is most practical and economical. -
Large and Small Gaps Between Consecutive Niven Numbers
1 2 Journal of Integer Sequences, Vol. 6 (2003), 3 Article 03.2.5 47 6 23 11 Large and small gaps between consecutive Niven numbers Jean-Marie De Koninck1 and Nicolas Doyon D¶epartement de math¶ematiques et de statistique Universit¶e Laval Qu¶ebec G1K 7P4 Canada [email protected] [email protected] Abstract A positive integer is said to be a Niven number if it is divisible by the sum of its decimal digits. We investigate the occurrence of large and small gaps between consecutive Niven numbers. 1 Introduction A positive integer n is said to be a Niven number (or a Harshad number) if it is divisible by the sum of its (decimal) digits. For instance, 153 is a Niven number since 9 divides 153, while 154 is not. Niven numbers have been extensively studied; see for instance Cai [1], Cooper and Kennedy [2], Grundman [5] or Vardi [6]. Let N(x) denote the number of Niven numbers · x. Recently, De Koninck and Doyon proved [3], using elementary methods, that given any " > 0, x log log x x1¡" ¿ N(x) ¿ : log x Later, using complex variables as well as probabilistic number theory, De Koninck, Doyon and K¶atai [4] showed that x N(x) = (c + o(1)) ; (1) log x 1Research supported in part by a grant from NSERC. 1 where c is given by 14 c = log 10 ¼ 1:1939: (2) 27 In this paper, we investigate the occurrence of large gaps between consecutive Niven numbers. Secondly, denoting by T (x) the number of Niven numbers n · x such that n + 1 is also a Niven number, we prove that x log log x T (x) ¿ : (log x)2 We conclude by stating a conjecture. -
~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. -
Generalized Sum of Stella Octangula Numbers
DOI: 10.5281/zenodo.4662348 Generalized Sum of Stella Octangula Numbers Amelia Carolina Sparavigna Department of Applied Science and Technology, Politecnico di Torino A generalized sum is an operation that combines two elements to obtain another element, generalizing the ordinary addition. Here we discuss that concerning the Stella Octangula Numbers. We will also show that the sequence of these numbers, OEIS A007588, is linked to sequences OEIS A033431, OEIS A002378 (oblong or pronic numbers) and OEIS A003154 (star numbers). The Cardano formula is also discussed. In fact, the sequence of the positive integers can be obtained by means of Cardano formula from the sequence of Stella Octangula numbers. Keywords: Groupoid Representations, Integer Sequences, Binary Operators, Generalized Sums, OEIS, On-Line Encyclopedia of Integer Sequences, Cardano formula. Torino, 5 April 2021. A generalized sum is a binary operation that combines two elements to obtain another element. In particular, this operation acts on a set in a manner that its two domains and its codomain are the same set. Some generalized sums have been previously proposed proposed for different sets of numbers (Fibonacci, Mersenne, Fermat, q-numbers, repunits and others). The approach was inspired by the generalized sums used for entropy [1,2]. The analyses of sequences of integers and q-numbers have been collected in [3]. Let us repeat here just one of these generalized sums, that concerning the Mersenne n numbers [4]. These numbers are given by: M n=2 −1 . The generalized sum is: M m⊕M n=M m+n=M m+M n+M m M n In particular: M n⊕M 1=M n +1=M n+M 1+ M n M 1 1 DOI: 10.5281/zenodo.4662348 The generalized sum is the binary operation which is using two Mersenne numbers to have another Mersenne number. -
Some Links of Balancing and Cobalancing Numbers with Pell and Associated Pell Numbers
Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 6 (2011), No. 1, pp. 41-72 SOME LINKS OF BALANCING AND COBALANCING NUMBERS WITH PELL AND ASSOCIATED PELL NUMBERS G. K. PANDA1,a AND PRASANTA KUMAR RAY2,b 1 National Institute of Technology, Rourkela -769 008, Orissa, India. a E-mail: gkpanda nit@rediffmail.com 2 College of Arts Science and Technology, Bandomunda, Rourkela -770 032, Orissa, India. b E-mail: [email protected] Abstract Links of balancing and cobalancing numbers with Pell and associated Pell numbers th th are established. It is proved that the n balancing number is product of the n Pell th number and the n associated Pell number. It is further observed that the sequences of balancing and cobalancing numbers are very closely related to the Pell sequence whereas, the sequences of Lucas-balancing and Lucas-cobalancing numbers constitute the associated Pell sequence. The solutions of some Diophantine equations including Pythagorean and Pythagorean-type equations are obtained in terms of these numbers. 1. Introduction The study of number sequences has been a source of attraction to the mathematicians since ancient times. From that time many mathematicians have been focusing their attention on the study of the fascinating triangu- lar numbers (numbers of the form n(n + 1)/2 where n Z+ are known as ∈ triangular numbers). Behera and Panda [1], while studying the Diophan- tine equation 1 + 2 + + (n 1) = (n + 1) + (n +2)+ + (n + r) on · · · − · · · triangular numbers, obtained an interesting relation of the numbers n in Received April 28, 2009 and in revised form September 25, 2009. -
Integer Sequences
UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular -
Fermat Pseudoprimes
1 TWO HUNDRED CONJECTURES AND ONE HUNDRED AND FIFTY OPEN PROBLEMS ON FERMAT PSEUDOPRIMES (COLLECTED PAPERS) Education Publishing 2013 Copyright 2013 by Marius Coman Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721 Peer-Reviewers: Dr. A. A. Salama, Faculty of Science, Port Said University, Egypt. Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Mohamed Eisa, Dept. of Computer Science, Port Said Univ., Egypt. EAN: 9781599732572 ISBN: 978-1-59973-257-2 1 INTRODUCTION Prime numbers have always fascinated mankind. For mathematicians, they are a kind of “black sheep” of the family of integers by their constant refusal to let themselves to be disciplined, ordered and understood. However, we have at hand a powerful tool, insufficiently investigated yet, which can help us in understanding them: Fermat pseudoprimes. It was a night of Easter, many years ago, when I rediscovered Fermat’s "little" theorem. Excited, I found the first few Fermat absolute pseudoprimes (561, 1105, 1729, 2465, 2821, 6601, 8911…) before I found out that these numbers are already known. Since then, the passion for study these numbers constantly accompanied me. Exceptions to the above mentioned theorem, Fermat pseudoprimes seem to be more malleable than prime numbers, more willing to let themselves to be ordered than them, and their depth study will shed light on many properties of the primes, because it seems natural to look for the rule studying it’s exceptions, as a virologist search for a cure for a virus studying the organisms that have immunity to the virus. -
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. -
PRONIC FIBONACCI NUMBERS Wayne L. Mcdanie!
PRONIC FIBONACCI NUMBERS Wayne L. McDanie! University of Missouri-St. Louis, St. Louis, MO 63121 (Submitted April 1996-Final Revision June 1996) 1. INTRODUCTION Tronic" is an old-fashioned term meaning "the product of two consecutive integers." (The reader will find the term indexed in [1], referring to some half-dozen articles.) In this paper we show that the only Fibonacci numbers that are the product of two consecutive integers are F0 - 0 andi^3 = 2. The referee of this paper has called the author's attention to the prior publication (December 1996) of this result in Chinese (see Ming Luo [3]). However, because of the relative inaccessi- bility of the earlier result, the referee recommended publication of this article in the Quarterly. If Fn = r(r +1), then 4Fn +1 is a square. Our approach is to show that Fn, for n'& 0, ±3, is not a pronic number by finding an integer w(ri) such that 4Fn +1 is a quadratic nonresidue modulo w(ri). There is a sense in which this paper may be considered a companion paper to Ming Luo's article on triangular numbers in the sequence of Fibonacci numbers: If Fn is a pronic number, then Fn is two times a triangular number. We shall use two results from Luo's paper, and take advan- tage of the periodicity of the sequence modulo an appropriate integer w(ri), enabling us to prove our result through use of the Jacobi symbol (4Fn +1 \w(n)) in a finite number of cases. Our main result is the following theorem. -
Properties of 2020
Properties of 2020 freeman66 February 2, 2020 Abstract This document provides some properties of 2020. If you have a good one, feel free to let me know! Also, note that the properties in each section go from easy to interesting to overkill. Note that the ones that look weird and/or are overkill are likely from OEIS, so I'm not weird OEIS is. 1 Algebra 1.1 Operations on 2020 p p 20202 is 4080400 and 20203 is 8242408000. 2020 is 44:9444101085 and 3 2020 = 12:6410686485. ln 2020 is 7:6108527903953 and log10 2020 is 3:3053513694466. sin 2020 is 0.044061988343923, 1 cos 2020 is -0.99902879897588, and tan 2020 is -0.044104822993183. 2020 has period 4. 1.2 Time 2020 seconds is equal to 33 minutes, 40 seconds. 2020 is a leap year. 2020 has 53 Wednesdays and Thursdays. The previous year this occurs was 1992, and the next is 2048. 1.3 Triangular Numbers P22 2020 is equal to k=3 Tk, where Tk is the kth triangular number. Also, T5 + T30 + T55 = 2020. 1.4 Heptagonal Numbers The alternating sum of the first 40 heptagonal numbers is 2020. 1.5 Sequences A sequence that starts with 4 then adds 1, then 2, then 3, and so on contains 2020. Also, in the 2 P1 xi i=1 2 1 k 14 1−xi product f(x) = Π (1 + 4 · x ), x has coefficient 2020. Also, in the expansion of 2 , k=1 1 xi Πi=1 2 1−xi 51 n the coefficient of x is 2020. -
Sum of the Reciprocals of Famous Series: Mathematical Connections with Some Sectors of Theoretical Physics and String Theory
1Torino, 14/04/2016 Sum of the reciprocals of famous series: mathematical connections with some sectors of theoretical physics and string theory 1,2 Ing. Pier Franz Roggero, Dr. Michele Nardelli , P.i. Francesco Di Noto 1Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy 2 Dipartimento di Matematica ed Applicazioni “R. Caccioppoli” Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy Abstract In this paper it has been calculated the sums of the reciprocals of famous series. The sum of the reciprocals gives fundamental information on these series. The higher this sum and larger numbers there are in series and vice versa. Furthermore we understand also what is the growth factor of the series and that there is a clear link between the sums of the reciprocal and the "intrinsic nature" of the series. We have described also some mathematical connections with some sectors of theoretical physics and string theory 2Torino, 14/04/2016 Index: 1. KEMPNER SERIES ........................................................................................................................................................ 3 2. SEXY PRIME NUMBERS .............................................................................................................................................. 6 3. TWIN PRIME NUMBERS .............................................................................................................................................