Prime Numbers and Their Analysis
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Number 73 Is the 37Th Odd Number
73 Seventy-Three LXXIII i ii iii iiiiiiiiii iiiiiiiii iiiiiiii iiiiiii iiiiiiii iiiiiiiii iiiiiiiiii iii ii i Analogous ordinal: seventy-third. The number 73 is the 37th odd number. Note the reversal here. The number 73 is the twenty-first prime number. The number 73 is the fifty-sixth deficient number. The number 73 is in the eighth twin-prime pair 71, 73. This is the last of the twin primes less than 100. No one knows if the list of twin primes ever ends. As a sum of four or fewer squares: 73 = 32 + 82 = 12 + 62 + 62 = 12 + 22 + 22 + 82 = 22 + 22 + 42 + 72 = 42 + 42 + 42 + 52. As a sum of nine or fewer cubes: 73 = 3 13 + 2 23 + 2 33 = 13 + 23 + 43. · · · As a difference of two squares: 73 = 372 362. The number 73 appears in two Pythagorean triples: [48, 55, 73] and [73, 2664, 2665]. Both are primitive, of course. As a sum of three odd primes: 73 = 3 + 3 + 67 = 3 + 11 + 59 = 3 + 17 + 53 = 3 + 23 + 47 = 3 + 29 + 41 = 5 + 7 + 61 = 5 + 31 + 37 = 7 + 7 + 59 = 7 + 13 + 53 = 7 + 19 + 47 = 7 + 23 + 43 = 7 + 29 + 37 = 11 + 19 + 43 = 11 + 31 + 31 = 13 + 13 + 47 = 13 + 17 + 43 = 13 + 19 + 41 = 13 + 23 + 37 = 13 + 29 + 31 = 17 + 19 + 37 = 19 + 23 + 31. The number 73 is the 21st prime number. It’s reversal, 37, is the 12th prime number. Notice that if you strip off the six triangular points from the 73-circle hexagram pictured above, you are left with a hexagon of 37 circles. -
Number World Number Game "Do You Like Number Games?", Zaina Teacher Asked
1 Number World Number game "Do you like number games?", Zaina teacher asked. "Oh! Yes!", said the children. "I'll say a number; you give me the next number at once. Ready?" "Ready!" "Ten", teacher began. "Eleven", said all the children. "Forty three" "Forty four" The game went on. "Four thousand ninety nine", teacher said. "Five thousand", replied some one. "Oh! No!... Four thousand and hundred", some caught on. Such mistakes are common. Try this on your friends. First Day Fiesta What is the number of children in First Day Fiesta class 1? What is the largest number you can read? What is the largest four-digit number? What is the next number? 435268 children in Class 1. 8 And the largest five-digit number? What is the next number? How do we find this number? Giant number How do we read it? Look at the table of large numbers: If we are asked for a large number, we often say crore or hundred crore. Put- 1 One ting ten zeros after one makes thou- 10 Ten sand crore. Think about the size of the number with hundred zeros after one. 100 Hundred This is called googol. This name was 1000 Thousand popularized by Edward Kasner in 10000 Ten thousand 1938. 100000 Lakh In most countries, one lakh is named hundred thousand and ten lakh is 1000000 Ten lakh named million. 10000000 Crore 100000000 Ten crore You're always counting This continues with hundred crore, thousand numbers! What's crore, and so on. your goal? Now can you say what we get when we add one Googol! to ninety nine thousand nine hundred and ninety nine? 99999 + 1 = 100000 How do we read this? Look it up in the table. -
Gaussian Prime Labeling of Super Subdivision of Star Graphs
of Math al em rn a u ti o c J s l A a Int. J. Math. And Appl., 8(4)(2020), 35{39 n n d o i i t t a s n A ISSN: 2347-1557 r e p t p n l I i c • Available Online: http://ijmaa.in/ a t 7 i o 5 n 5 • s 1 - 7 4 I 3 S 2 S : N International Journal of Mathematics And its Applications Gaussian Prime Labeling of Super Subdivision of Star Graphs T. J. Rajesh Kumar1,∗ and Antony Sanoj Jerome2 1 Department of Mathematics, T.K.M College of Engineering, Kollam, Kerala, India. 2 Research Scholar, University College, Thiruvananthapuram, Kerala, India. Abstract: Gaussian integers are the complex numbers of the form a + bi where a; b 2 Z and i2 = −1 and it is denoted by Z[i]. A Gaussian prime labeling on G is a bijection from the vertices of G to [ n], the set of the first n Gaussian integers in the spiral ordering such that if uv 2 E(G), then (u) and (v) are relatively prime. Using the order on the Gaussian integers, we discuss the Gaussian prime labeling of super subdivision of star graphs. MSC: 05C78. Keywords: Gaussian Integers, Gaussian Prime Labeling, Super Subdivision of Graphs. © JS Publication. 1. Introduction The graphs considered in this paper are finite and simple. The terms which are not defined here can be referred from Gallian [1] and West [2]. A labeling or valuation of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy, a label depending upon the vertex labels f(x) and f(y). -
Mathematical Circus & 'Martin Gardner
MARTIN GARDNE MATHEMATICAL ;MATH EMATICAL ASSOCIATION J OF AMERICA MATHEMATICAL CIRCUS & 'MARTIN GARDNER THE MATHEMATICAL ASSOCIATION OF AMERICA Washington, DC 1992 MATHEMATICAL More Puzzles, Games, Paradoxes, and Other Mathematical Entertainments from Scientific American with a Preface by Donald Knuth, A Postscript, from the Author, and a new Bibliography by Mr. Gardner, Thoughts from Readers, and 105 Drawings and Published in the United States of America by The Mathematical Association of America Copyright O 1968,1969,1970,1971,1979,1981,1992by Martin Gardner. All riglhts reserved under International and Pan-American Copyright Conventions. An MAA Spectrum book This book was updated and revised from the 1981 edition published by Vantage Books, New York. Most of this book originally appeared in slightly different form in Scientific American. Library of Congress Catalog Card Number 92-060996 ISBN 0-88385-506-2 Manufactured in the United States of America For Donald E. Knuth, extraordinary mathematician, computer scientist, writer, musician, humorist, recreational math buff, and much more SPECTRUM SERIES Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications ANDREW STERRETT, JR.,Chairman Spectrum Editorial Board ROGER HORN, Chairman SABRA ANDERSON BART BRADEN UNDERWOOD DUDLEY HUGH M. EDGAR JEANNE LADUKE LESTER H. LANGE MARY PARKER MPP.a (@ SPECTRUM Also by Martin Gardner from The Mathematical Association of America 1529 Eighteenth Street, N.W. Washington, D. C. 20036 (202) 387- 5200 Riddles of the Sphinx and Other Mathematical Puzzle Tales Mathematical Carnival Mathematical Magic Show Contents Preface xi .. Introduction Xlll 1. Optical Illusions 3 Answers on page 14 2. Matches 16 Answers on page 27 3. -
On Terms of Generalized Fibonacci Sequences Which Are Powers of Their Indexes
mathematics Article On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes Pavel Trojovský Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic, [email protected]; Tel.: +42-049-333-2860 Received: 29 June 2019; Accepted: 31 July 2019; Published: 3 August 2019 (k) Abstract: The k-generalized Fibonacci sequence (Fn )n (sometimes also called k-bonacci or k-step Fibonacci sequence), with k ≥ 2, is defined by the values 0, 0, ... , 0, 1 of starting k its terms and such way that each term afterwards is the sum of the k preceding terms. This paper is devoted to the proof of the fact that the (k) t (2) 2 (3) 2 Diophantine equation Fm = m , with t > 1 and m > k + 1, has only solutions F12 = 12 and F9 = 9 . Keywords: k-generalized Fibonacci sequence; Diophantine equation; linear form in logarithms; continued fraction MSC: Primary 11J86; Secondary 11B39. 1. Introduction The well-known Fibonacci sequence (Fn)n≥0 is given by the following recurrence of the second order Fn+2 = Fn+1 + Fn, for n ≥ 0, with the initial terms F0 = 0 and F1 = 1. Fibonacci numbers have a lot of very interesting properties (see e.g., book of Koshy [1]). One of the famous classical problems, which has attracted a attention of many mathematicians during the last thirty years of the twenty century, was the problem of finding perfect powers in the sequence of Fibonacci numbers. Finally in 2006 Bugeaud et al. [2] (Theorem 1), confirmed these expectations, as they showed that 0, 1, 8 and 144 are the only perfect powers in the sequence of Fibonacci numbers. -
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000)
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000) 1. INTRODUCTION In 1982, Albert Wilansky, a mathematics professor at Lehigh University wrote a short article in the Two-Year College Mathematics Journal [6]. In that article he identified a new subset of the composite numbers. He defined a Smith number to be a composite number where the sum of the digits in its prime factorization is equal to the digit sum of the number. The set was named in honor of Wi!anskyJs brother-in-law, Dr. Harold Smith, whose telephone number 493-7775 when written as a single number 4,937,775 possessed this interesting characteristic. Adding the digits in the number and the digits of its prime factors 3, 5, 5 and 65,837 resulted in identical sums of42. Wilansky provided two other examples of numbers with this characteristic: 9,985 and 6,036. Since that time, many things have been discovered about Smith numbers including the fact that there are infinitely many Smith numbers [4]. The largest Smith numbers were produced by Samuel Yates. Using a large repunit and large palindromic prime, Yates was able to produce Smith numbers having ten million digits and thirteen million digits. Using the same large repunit and a new large palindromic prime, the author is able to find a Smith number with over thirty-two million digits. 2. NOTATIONS AND BASIC FACTS For any positive integer w, we let S(ri) denote the sum of the digits of n. -
Crazy Representations of Natural Numbers, Selfie Numbers
Inder J. Taneja RGMIA Research Report Collection, 19(2016), pp.1-37, http://rgmia.org/v19.php Crazy Representations of Natural Numbers, Selfie Numbers, Fibonacci Sequence, and Selfie Fractions Inder J. Taneja1 SUMMARY This summary brings author’s work on numbers. The study is made in different ways. Specially, towards, Crazy Representations of Natural Numbers, Running Expressions, Selfie Numbers etc. Natural numbers are represented in different situations, such as, writing in terms of 1 to 9 or reverse, flexible power of same digits as bases, single digit, single letter, etc. Expressions appearing with equalities having 1 to 9 or 9 to 1 or 9 to 0, calling running expressions are also presented. In continuation, there is work is on selfie numbers, unified, patterns, symmetrical representations in selfie numbers, Fibonacci sequence and selfie numbers, flexible power Narcissistic and selfie numbers, selfie fractions, etc. The selfie numbers may also be considered as generalized or wild narcissistic numbers, where natural numbers are represented by their own digits with certain operations. The study is also made towards equivalent fractions and palindromic-type numbers. This summary is separated by sections and subsections given as follows: 1 Crazy Representations of Natural Numbers [1]; 2 Flexible Power Representations [34]; 2.1 Unequal String Lengths [33]; 2.2 Equal String Lengths [25]; 3 Pyramidical Representations [20, 24, 31, 32]; 3.1 Crazy Representations [33]; 3.2 Flexible Power [25]; 4 Double Sequential Representations [20, 24, 31, 32]; 5 Triple Sequential Representations; [35]; 6 Single Digit Representations; [2]; 7 Single Letter Representations [4, 8]; 7.1 Single Letter Power Representations [8]; 7.2 Palindromic and Number Patterns [9, 10]; 8 Running Expressions. -
Number Systems and Radix Conversion
Number Systems and Radix Conversion Sanjay Rajopadhye, Colorado State University 1 Introduction These notes for CS 270 describe polynomial number systems. The material is not in the textbook, but will be required for PA1. We humans are comfortable with numbers in the decimal system where each po- sition has a weight which is a power of 10: units have a weight of 1 (100), ten’s have 10 (101), etc. Even the fractional apart after the decimal point have a weight that is a (negative) power of ten. So the number 143.25 has a value that is 1 ∗ 102 + 4 ∗ 101 + 3 ∗ 0 −1 −2 2 5 10 + 2 ∗ 10 + 5 ∗ 10 , i.e., 100 + 40 + 3 + 10 + 100 . You can think of this as a polyno- mial with coefficients 1, 4, 3, 2, and 5 (i.e., the polynomial 1x2 + 4x + 3 + 2x−1 + 5x−2+ evaluated at x = 10. There is nothing special about 10 (just that humans evolved with ten fingers). We can use any radix, r, and write a number system with digits that range from 0 to r − 1. If our radix is larger than 10, we will need to invent “new digits”. We will use the letters of the alphabet: the digit A represents 10, B is 11, K is 20, etc. 2 What is the value of a radix-r number? Mathematically, a sequence of digits (the dot to the right of d0 is called the radix point rather than the decimal point), : : : d2d1d0:d−1d−2 ::: represents a number x defined as follows X i x = dir i 2 1 0 −1 −2 = ::: + d2r + d1r + d0r + d−1r + d−2r + ::: 1 Example What is 3 in radix 3? Answer: 0.1. -
On the First Occurrences of Gaps Between Primes in a Residue Class
On the First Occurrences of Gaps Between Primes in a Residue Class Alexei Kourbatov JavaScripter.net Redmond, WA 98052 USA [email protected] Marek Wolf Faculty of Mathematics and Natural Sciences Cardinal Stefan Wyszy´nski University Warsaw, PL-01-938 Poland [email protected] To the memory of Professor Thomas R. Nicely (1943–2019) Abstract We study the first occurrences of gaps between primes in the arithmetic progression (P): r, r + q, r + 2q, r + 3q,..., where q and r are coprime integers, q > r 1. ≥ The growth trend and distribution of the first-occurrence gap sizes are similar to those arXiv:2002.02115v4 [math.NT] 20 Oct 2020 of maximal gaps between primes in (P). The histograms of first-occurrence gap sizes, after appropriate rescaling, are well approximated by the Gumbel extreme value dis- tribution. Computations suggest that first-occurrence gaps are much more numerous than maximal gaps: there are O(log2 x) first-occurrence gaps between primes in (P) below x, while the number of maximal gaps is only O(log x). We explore the con- nection between the asymptotic density of gaps of a given size and the corresponding generalization of Brun’s constant. For the first occurrence of gap d in (P), we expect the end-of-gap prime p √d exp( d/ϕ(q)) infinitely often. Finally, we study the gap ≍ size as a function of its index in thep sequence of first-occurrence gaps. 1 1 Introduction Let pn be the n-th prime number, and consider the difference between successive primes, called a prime gap: pn+1 pn. -
Sequences of Primes Obtained by the Method of Concatenation
SEQUENCES OF PRIMES OBTAINED BY THE METHOD OF CONCATENATION (COLLECTED PAPERS) Copyright 2016 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: 9781599734668 ISBN: 978-1-59973-466-8 1 INTRODUCTION The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two numbers by their numerals. That is, the concatenation of 69 and 420 is 69420”. Though the method of concatenation is widely considered as a part of so called “recreational mathematics”, in fact this method can often lead to very “serious” results, and even more than that, to really amazing results. This is the purpose of this book: to show that this method, unfairly neglected, can be a powerful tool in number theory. In particular, as revealed by the title, I used the method of concatenation in this book to obtain possible infinite sequences of primes. Part One of this book, “Primes in Smarandache concatenated sequences and Smarandache-Coman sequences”, contains 12 papers on various sequences of primes that are distinguished among the terms of the well known Smarandache concatenated sequences (as, for instance, the prime terms in Smarandache concatenated odd -
Number Theory Course Notes for MA 341, Spring 2018
Number Theory Course notes for MA 341, Spring 2018 Jared Weinstein May 2, 2018 Contents 1 Basic properties of the integers 3 1.1 Definitions: Z and Q .......................3 1.2 The well-ordering principle . .5 1.3 The division algorithm . .5 1.4 Running times . .6 1.5 The Euclidean algorithm . .8 1.6 The extended Euclidean algorithm . 10 1.7 Exercises due February 2. 11 2 The unique factorization theorem 12 2.1 Factorization into primes . 12 2.2 The proof that prime factorization is unique . 13 2.3 Valuations . 13 2.4 The rational root theorem . 15 2.5 Pythagorean triples . 16 2.6 Exercises due February 9 . 17 3 Congruences 17 3.1 Definition and basic properties . 17 3.2 Solving Linear Congruences . 18 3.3 The Chinese Remainder Theorem . 19 3.4 Modular Exponentiation . 20 3.5 Exercises due February 16 . 21 1 4 Units modulo m: Fermat's theorem and Euler's theorem 22 4.1 Units . 22 4.2 Powers modulo m ......................... 23 4.3 Fermat's theorem . 24 4.4 The φ function . 25 4.5 Euler's theorem . 26 4.6 Exercises due February 23 . 27 5 Orders and primitive elements 27 5.1 Basic properties of the function ordm .............. 27 5.2 Primitive roots . 28 5.3 The discrete logarithm . 30 5.4 Existence of primitive roots for a prime modulus . 30 5.5 Exercises due March 2 . 32 6 Some cryptographic applications 33 6.1 The basic problem of cryptography . 33 6.2 Ciphers, keys, and one-time pads . -
Ramanujan, Robin, Highly Composite Numbers, and the Riemann Hypothesis
Contemporary Mathematics Volume 627, 2014 http://dx.doi.org/10.1090/conm/627/12539 Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis Jean-Louis Nicolas and Jonathan Sondow Abstract. We provide an historical account of equivalent conditions for the Riemann Hypothesis arising from the work of Ramanujan and, later, Guy Robin on generalized highly composite numbers. The first part of the paper is on the mathematical background of our subject. The second part is on its history, which includes several surprises. 1. Mathematical Background Definition. The sum-of-divisors function σ is defined by 1 σ(n):= d = n . d d|n d|n In 1913, Gr¨onwall found the maximal order of σ. Gr¨onwall’s Theorem [8]. The function σ(n) G(n):= (n>1) n log log n satisfies lim sup G(n)=eγ =1.78107 ... , n→∞ where 1 1 γ := lim 1+ + ···+ − log n =0.57721 ... n→∞ 2 n is the Euler-Mascheroni constant. Gr¨onwall’s proof uses: Mertens’s Theorem [10]. If p denotes a prime, then − 1 1 1 lim 1 − = eγ . x→∞ log x p p≤x 2010 Mathematics Subject Classification. Primary 01A60, 11M26, 11A25. Key words and phrases. Riemann Hypothesis, Ramanujan’s Theorem, Robin’s Theorem, sum-of-divisors function, highly composite number, superabundant, colossally abundant, Euler phi function. ©2014 American Mathematical Society 145 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 146 JEAN-LOUIS NICOLAS AND JONATHAN SONDOW Figure 1. Thomas Hakon GRONWALL¨ (1877–1932) Figure 2. Franz MERTENS (1840–1927) Nowwecometo: Ramanujan’s Theorem [2, 15, 16].