DOCSLIB.ORG

Generalized Partitions and New Ideas on Number Theory and Smarandache Sequences

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Generalized Partitions and New Ideas on Number Theory and Smarandache Sequences Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences Editor’s Note This book arose out of a collection of papers written by Amarnath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man who seems to have no end of ideas. Most of the papers were published in Smarandache Notions Journal and there was a great deal of overlap. My intent in transforming the papers into a coherent book was to remove the duplications, organize the material based on topic and clean up some of the most obvious errors. However, I made no attempt to verify every statement, so the mathematical work is almost exclusively that of Murthy. I would also like to thank Tyler Brogla, who created the image that appears on the front cover. Charles Ashbacher Smarandache Repeatable Reciprocal Partition of Unity {2, 3, 10, 15} 1/2 + 1/3 + 1/10 + 1/15 = 1 Amarnath Murthy / Charles Ashbacher AMARNATH MURTHY S.E.(E&T) WELL LOGGING SERVICES OIL AND NATURAL GAS CORPORATION LTD CHANDKHEDA AHMEDABAD GUJARAT- 380005 INDIA CHARLES ASHBACHER MOUNT MERCY COLLEGE 1330 ELMHURST DRIVE NE CEDAR RAPIDS, IOWA 42402 USA GENERALIZED PARTITIONS AND SOME NEW IDEAS ON NUMBER THEORY AND SMARANDACHE SEQUENCES Hexis Phoenix 2005 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) http://wwwlib.umi.com/bod/search/basic Peer Reviewers: 1) Eng. Marian Popescu, Str. Spaniei, Bl. O8, Sc. 1, Ap. 14, Craiova, Jud. Dolj, Romania. 2) Dr. Sukanto Bhattacharya, Department of Business Administration, Alaska Pacific University, U.S.A. 3) Dr. M. Khoshnevisan, School of Accounting and Finance, Griffith University, Gold Coast, Queensland 9726, Australia. Copyright 2005 by Hexis., Amarnath Murthy and Charles Ashbacher Many books can be downloaded from the following E-Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm Front cover image © Tyler Brogla and Charles Ashbacher ISBN: 1-931233-34-9 Standard Address Number: 297-5092 Printed in the United States of America 2 Editor’s Note This book arose out of a collection of papers written by Amarnath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man who seems to have no end of ideas. Most of the papers were published in Smarandache Notions Journal and there was a great deal of overlap. My intent in transforming the papers into a coherent book was to remove the duplications, organize the material based on topic and clean up some of the most obvious errors. However, I made no attempt to verify every statement, so the mathematical work is almost exclusively that of Murthy. I would also like to thank Tyler Brogla, who created the image that appears on the front cover. Charles Ashbacher 3 Table of Contents Editor’s Note ………………….. …………………… 3 Table of Contents …………………. …………………… 4 Chapter 1 Smarandache ………………. 11 Partition Functions 1.1 Smarandache ………………. 11 Partition Sets, Sequences and Functions 1.2 A Program to ………………. 16 Determine the Number of Smarandache Distinct Reciprocal Partitions of Unity of a Given Length 1.3 A Note On Maohua ………………. 18 Le’s Proof of Murthy’s Conjecture On Reciprocal Partition Theory 1.4 Generalization of ………………. 19 Partition Functions, Introduction of the Smarandache Partition Function 1.5 Open Problems and ………………. 28 Conjectures On the Factor/Reciprocal Partition Theory 1.6 A General Result on ………………. 30 the Smarandache Star Function 1.7 More Results and ………………. 40 Applications of the Generalized Smarandache Star Function 1.8 Properties of the ……………….. 47 Smarandache Star Triangle 1.9 Smarandache Factor ……………….. 50 4 Partitions Of A Typical Canonical Form 1.10 Length/Extent of ……………….. 55 Smarandache Factor Partitions 1.11 More Ideas On ………………… 58 Smarandache Factor Partitions 1.12 A Note On the ………………… 61 Smarandache Divisor Sequences 1.13 An Algorithm for ………………… 62 Listing the Smarandache Factor Partitions 1.14 A Program For ………………… 64 Determining the Number of SFPs 1.15 Chapter References ………………… 70 Chapter 2 Smarandache ………………… 72 Sequences 2.1 On the Largest Balu ………………… 72 Number and Some SFP Equations 2.2 Smarandache Pascal ………………… 75 Derived Sequences 2.3 Depascalization of ………………… 81 Smarandache Pascal Derived Sequences and Backward Extended Fibonacci Sequence 2.4 Proof the the ………………… 83 Depascalization Theorem 2.5 Smarandache ………………… 86 Friendly Numbers and A Few More Sequences 2.6 Some New ………………… 89 Smarandache Sequences, Functions and Partitions 5 2.7 Smarandache ………………… 94 Reverse Auto Correlated Sequences and Some Fibonacci Derived Smarandache Sequences 2.8 Smarandache Star ………………… 98 (Stirling) Derived Sequences 2.9 Smarandache ………………… 100 Strictly Staircase Sequence 2.10 The Sum of the ………………… 101 Reciprocals of the Smarandache Multiplicative Sequence 2.11 Decomposition of ………………… 103 the Divisors of A Natural Number Into Pairwise Co- Prime Sets 2.12 On the Divisors of ………………… 106 the Smarandache Unary Sequence 2.13 Smarandache Dual ………………… 108 Symmetric Functions and Corresponding Numbers of the Type of Stirling Numbers of the First Kind 2.14 On the Infinitude of ………………… 110 the Smarandache Additive Square Sequence 2.15 On the Infinitude of ………………… 111 the Smarandache Multiplicative Square Sequence 2.16 Another ………………… 113 Classification of the Ocean of 6 Smarandache Sequences 2.17 Pouring a Few ………………… 114 Drops in the Ocean of Smarandache Sequences and Series 2.18 Smarandache …………………. 118 Pythagoras Additive Square Sequence 2.19 The Number of …………………. 120 Elements the Smarandache Multiplicative Square Sequence and the Smarandache Additive Square Sequence Have in Common 2.20 Smarandache ………………… 121 Patterned Perfect Cube Sequences 2.21 The Smarandache ………………… 122 Additive Cube Sequence is Infinite 2.22 More Examples and ………………… 123 Results On the Infinitude of Certain Smarandache Sequences 2.23 Smarandache ………………… 124 Symmetric (Palindromic) Perfect Power Sequences 2.24 Some Propositions ………………… 126 On the Smarandache n2n Sequence 2.25 The Smarandache ………………… 128 Fermat Additive Cube Sequence 2.26 The Smarandache ………………… 130 nn2 Sequence Contains No Perfect 7 Squares 2.27 Primes in the ………………… 132 Smarandache nnm Sequence 2.28 Some Ideas On the ………………….. 133 Smarandache nkn Sequence 2.29 Some Notions On ………………….. 134 Least Common Multiples 2.30 An Application of ………………….. 137 the Smarandache LCM Sequence and the Largest Number Divisible by All the Integers Not Exceeding Its rth Root 2.31 The Number of ………………….. 138 Primes In the Smarandache Multiple Sequence 2.32 More On the ………………….. 139 Smarandache Square and Higher Power Bases 2.33 Smarandache Fourth ………………….. 140 and Higher Patterned/Additive Perfect Power Sequences 2.34 The Smarandache ………………….. 142 Multiplicative Cubic Sequence and More Ideas on Digit Sums 2.35 Smarandache Prime ………………….. 143 Generator Sequence 2.36 Chapter References 145 Chapter 3 Miscellaneous …………………. 147 Topics 3.1 Exploring Some …………………. 147 New Ideas On Smarandache Type Sets, Functions and Sequences 8 3.2 Fabricating Perfect ………………….. 155 Squares With a Given Valid Digit Sum 3.3 Fabricating Perfect …………………… 157 Cubes With a Given Valid Digit Sum 3.4 Smarandache …………………… 161 Perfect Powers With Given Valid Digit Sum 3.5 Numbers That Are a …………………… 164 Multiple of the Product of Their Digits and Related Ideas 3.6 The Largest and ………………….. 166 Smallest mth Power Whose Digit Sum/Product Is It’s mth Root 3.7 A Conjecture on …………………. 168 d(N), the Divisor Function Itself As A Divisor With Required Justification 3.8 Smarandache …………………. 171 Fitorial and Supplementary Fitorial Functions 3.9 Some More …………………. 174 Conjectures On Primes and Divisors 3.10 Smarandache 176 Reciprocal Function and An Elementary Inequality 3.11 Smarandache …………………. 178 Maximum Reciprocal Representation Function 3.12 Smarandache …………………. 179 Determinant Sequence 9 3.13 Expansion of xn in …………………. 182 Smarandache Terms of Permutations 3.14 Miscellaneous …………………. 187 Results and Theorems on Smarandache Terms and Factor Partitions 3.15 Smarandache- ………………… 192 Murthy’s Figures of Periodic Symmetry of Rotation Specific to an Angle 3.16 Smarandache Route …………………. 198 Sequences 3.17 Smarandache …………………. 200 Geometrical Partitions and Sequences 3.18 Smarandache Lucky …………………. 207 Methods in Algebra, Trigonometry and Calculus 3.19 Chapter References …………………. 210 Index 212 10 Chapter 1 Smarandache Partition Functions Section 1 Smarandache Partition Sets, Sequences and Functions Unit fractions are fractions where the numerator is 1 and the denominator is a natural number. Our first point of interest is in determining all sets of unit fractions of a certain size where the sum of the elements in the set is 1. Definition: For n > 0, the Smarandache Repeatable Reciprocal partition of unity for n (SRRPS(n)) is the set of all sets of n natural numbers such that the sum of the reciprocals is 1. More formally, n SRRPS(n) = { x | x =(a1, a2, . ,an) where Σ 1/ar = 1 }. r=1 fRP(n) = order of the set SRRPS(n). For example, SRRPS(1) = { (1) }, fRP(1) = 1. SRRPS(2) = { (2,2)}, fRP(2) = 1. (1/2 + 1/2 = 1). SRRPS(3) = { (3,3,3), (2,3,6), (2,4,4)}, fRP(3) = 3. SRRPS(4) = { (4,4,4,4), (2,4,6,12),(2,3,7,42),(2,4,5,20),(2,6,6,6),(2,4,8,8), (2,3,12,12),(4,4,3,6),(3,3,6,6),(2,3,10,15),(2,3,9,18)}, fRP(4) = 14. Definition: The Smarandache Repeatable Reciprocal Partition of Unity Sequence is the sequence of numbers SRRPS(1), SRRPS(2), SRRPS(3), SRRPS(4), SRRPS(5), . Definition: For n > 0, the Smarandache Distinct Reciprocal Partition of Unity Set (SDRPS(n)) is SRRPS(n) where the elements of each set of size n must be unique. More formally, n SRRPS(n) = { x | x =(a1, a2, . ,an) where Σ 1/ar = 1 and ai = aj <=> i = j }.
Load more

Recommended publications
  • Ces`Aro's Integral Formula for the Bell Numbers (Corrected)
    Ces`aro’s Integral Formula for the Bell Numbers (Corrected) DAVID CALLAN Department of Statistics University of Wisconsin-Madison Medical Science Center 1300 University Ave Madison, WI 53706-1532 [email protected] October 3, 2005 In 1885, Ces`aro [1] gave the remarkable formula π 2 cos θ N = ee cos(sin θ)) sin( ecos θ sin(sin θ) ) sin pθ dθ p πe Z0 where (Np)p≥1 = (1, 2, 5, 15, 52, 203,...) are the modern-day Bell numbers. This formula was reproduced verbatim in the Editorial Comment on a 1941 Monthly problem [2] (the notation Np for Bell number was still in use then). I have not seen it in recent works and, while it’s not very profound, I think it deserves to be better known. Unfortunately, it contains a typographical error: a factor of p! is omitted. The correct formula, with n in place of p and using Bn for Bell number, is π 2 n! cos θ B = ee cos(sin θ)) sin( ecos θ sin(sin θ) ) sin nθ dθ n ≥ 1. n πe Z0 eiθ The integrand is the imaginary part of ee sin nθ, and so an equivalent formula is π 2 n! eiθ B = Im ee sin nθ dθ . (1) n πe Z0 The formula (1) is quite simple to prove modulo a few standard facts about set par- n titions. Recall that the Stirling partition number k is the number of partitions of n n [n] = {1, 2,...,n} into k nonempty blocks and the Bell number Bn = k=1 k counts n k n n k k all partitions of [ ].
    [Show full text]
  • Abstract in This Paper, D-Strong and Almost D-Strong Near-Rings Ha
    Periodica Mathematica Hungarlca Vol. 17 (1), (1986), pp. 13--20 D-STRONG AND ALMOST D-STRONG NEAR-RINGS A. K. GOYAL (Udaipur) Abstract In this paper, D-strong and almost D-strong near-rings have been defined. It has been proved that if R is a D-strong S-near ring, then prime ideals, strictly prime ideals and completely prime ideals coincide. Also if R is a D-strong near-ring with iden- tity, then every maximal right ideal becomes a maximal ideal and moreover every 2- primitive near-ring becomes a near-field. Several properties, chain conditions and structure theorems have also been discussed. Introduction In this paper, we have generalized some of the results obtained for rings by Wong [12]. Corresponding to the prime and strictly prime ideals in near-rings, we have defined D-strong and almost D-strong near-rings. It has been shown that a regular near-ring having all idempotents central in R is a D-strong and hence almost D-strong near-ring. If R is a D-strong S-near ring, then it has been shown that prime ideals, strictly prime ideals and com- pletely prime ideals coincide and g(R) = H(R) ~ ~(R), where H(R) is the intersection of all strictly prime ideals of R. Also if R is a D-strong near-ring with identity, then every maximal right ideal becomes a maximal ideal and moreover every 2-primitive near-ring becomes a near-field. Some structure theorems have also been discussed. Preliminaries Throughout R will denote a zero-symmetric left near-ring, i.e., R -- Ro in the sense of Pilz [10].
    [Show full text]
  • Neural Networks and the Search for a Quadratic Residue Detector
    Neural Networks and the Search for a Quadratic Residue Detector Michael Potter∗ Leon Reznik∗ Stanisław Radziszowski∗ [email protected] [email protected] [email protected] ∗ Department of Computer Science Rochester Institute of Technology Rochester, NY 14623 Abstract—This paper investigates the feasibility of employing This paper investigates the possibility of employing ANNs artificial neural network techniques for solving fundamental in the search for a solution to the quadratic residues (QR) cryptography problems, taking quadratic residue detection problem, which is defined as follows: as an example. The problem of quadratic residue detection Suppose that you have two integers a and b. Then a is is one which is well known in both number theory and a quadratic residue of b if and only if the following two cryptography. While it garners less attention than problems conditions hold: [1] such as factoring or discrete logarithms, it is similar in both 1) The greatest common divisor of a and b is 1, and difficulty and importance. No polynomial–time algorithm is 2) There exists an integer c such that c2 ≡ a (mod b). currently known to the public by which the quadratic residue status of one number modulo another may be determined. This The first condition is satisfied automatically so long as ∗ work leverages machine learning algorithms in an attempt to a 2 Zb . There is, however, no known polynomial–time (in create a detector capable of solving instances of the problem the number of bits of a and b) algorithm for determining more efficiently. A variety of neural networks, currently at the whether the second condition is satisfied.
    [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]
  • On Sufficient Conditions for the Existence of Twin Values in Sieves
    On Sufficient Conditions for the Existence of Twin Values in Sieves over the Natural Numbers by Luke Szramowski Submitted in Partial Fulfillment of the Requirements For the Degree of Masters of Science in the Mathematics Program Youngstown State University May, 2020 On Sufficient Conditions for the Existence of Twin Values in Sieves over the Natural Numbers Luke Szramowski I hereby release this thesis to the public. I understand that this thesis will be made available from the OhioLINK ETD Center and the Maag Library Circulation Desk for public access. I also authorize the University or other individuals to make copies of this thesis as needed for scholarly research. Signature: Luke Szramowski, Student Date Approvals: Dr. Eric Wingler, Thesis Advisor Date Dr. Thomas Wakefield, Committee Member Date Dr. Thomas Madsen, Committee Member Date Dr. Salvador A. Sanders, Dean of Graduate Studies Date Abstract For many years, a major question in sieve theory has been determining whether or not a sieve produces infinitely many values which are exactly two apart. In this paper, we will discuss a new result in sieve theory, which will give sufficient conditions for the existence of values which are exactly two apart. We will also show a direct application of this theorem on an existing sieve as well as detailing attempts to apply the theorem to the Sieve of Eratosthenes. iii Contents 1 Introduction 1 2 Preliminary Material 1 3 Sieves 5 3.1 The Sieve of Eratosthenes . 5 3.2 The Block Sieve . 9 3.3 The Finite Block Sieve . 12 3.4 The Sieve of Joseph Flavius .
    [Show full text]
  • MAT344 Lecture 6
    MAT344 Lecture 6 2019/May/22 1 Announcements 2 This week This week, we are talking about 1. Recursion 2. Induction 3 Recap Last time we talked about 1. Recursion 4 Fibonacci numbers The famous Fibonacci sequence starts like this: 1; 1; 2; 3; 5; 8; 13;::: The rule defining the sequence is F1 = 1;F2 = 1, and for n ≥ 3, Fn = Fn−1 + Fn−2: This is a recursive formula. As you might expect, if certain kinds of numbers have a name, they answer many counting problems. Exercise 4.1 (Example 3.2 in [KT17]). Show that a 2 × n checkerboard can be tiled with 2 × 1 dominoes in Fn+1 many ways. Solution: Denote the number of tilings of a 2 × n rectangle by Tn. We check that T1 = 1 and T2 = 2. We want to prove that they satisfy the recurrence relation Tn = Tn−1 + Tn−2: Consider the domino occupying the rightmost spot in the top row of the tiling. It is either a vertical domino, in which case the rest of the tiling can be interpreted as a tiling of a 2 × (n − 1) rectangle, or it is a horizontal domino, in which case there must be another horizontal domino under it, and the rest of the tiling can be interpreted as a tiling of a 2 × (n − 2) rectangle. Therefore Tn = Tn−1 + Tn−2: Since the number of tilings satisfies the same recurrence relation as the Fibonacci numbers, and T1 = F2 = 1 and T2 = F3 = 2, we may conclude that Tn = Fn+1.
    [Show full text]
  • 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.
    [Show full text]
  • 2019 Practice Mathcounts Solutions
    2019 Practice Mathcounts Solutions Austin Math Circle January 20, 2019 1 Sprint Round Problem 1. What is the sum of the first five odd numbers and the first four even numbers? Solution. This is simply the sum of all the integers one through nine, which is 45 . Proposed by Jay Leeds. Problem 2. Shipping a box costs a flat rate of $5 plus $2 for every pound after the first five pounds. How much does it cost to ship a 18-pound box? Solution. An 18-pound box has 13 excess pounds, so our fee is 13 $2 $5 $31 . ¢ Å Æ Proposed by Jay Leeds. Problem 3. In rectangle ABCD, AB 6, BC 8, and M is the midpoint of AB. What is the area of triangle Æ Æ CDM? Solution. We see that this triangle has base six and height eight, so its area is 6 8/2 24 . ¢ Æ Proposed by Jay Leeds. Problem 4. Creed flips three coins. What is the probability that he flips heads at least once? Express your answer as a common fraction. 3 Solution. There is a ¡ 1 ¢ 1 chance that Creed flips no heads, so the probability that he flips at least once heads 2 Æ 8 7 is 1 1 . ¡ 8 Æ 8 Proposed by Jay Leeds. Problem 5. Compute the median of the following five numbers: A 43 , B 4.5, C 23 , D 22, and E 4.9. Æ 9 Æ Æ 5 Æ Æ Write A, B, C, D, or E as your answer. Solution. We can easily sort 22 4, 4.5, and 4.9.
    [Show full text]
  • What Can Be Said About the Number 13 Beyond the Fact That It Is a Prime Number? *
    CROATICA CHEMICA ACTA CCACAA 77 (3) 447¿456 (2004) ISSN-0011-1643 CCA-2946 Essay What Can Be Said about the Number 13 beyond the Fact that It Is a Prime Number? * Lionello Pogliani,a Milan Randi},b and Nenad Trinajsti}c,** aDipartimento di Chimica, Universitá della Calabria, 879030 Rende (CS), Italy bDepartment of Mathematics and Computer Science, Drake University, Des Moines, Iowa 50311, USA cThe Rugjer Bo{kovi} Institute, P.O. Box 180, HR-10002 Zagreb, Croatia RECEIVED DECEMBER 2, 2003; REVISED MARCH 8, 2004; ACCEPTED MARCH 9, 2004 Key words The story of the number 13 that goes back to ancient Egypt is told. The mathematical signifi- art cance of 13 is briefly reviewed and 13 is discussed in various contexts, with special reference chemistry to the belief of many that this number is a rather unlucky number. Contrary examples are also history presented. Regarding everything, the number 13 appears to be a number that leaves no one in- literature different. number theory poetry science INTRODUCTION – religious [3 in the Christian faith, 5 in Islam]; – dramatic [3 and 9 by William Shakespeare (1564– »Numero pondere et mensura Deus omnia condidit.« 1616) in Macbeth]; Isaac Newton in 1722 [ 1 – literary 3 and 10 by Graham Greene (1904–1991) (Rozsondai and Rozsondai) in the titles of his successful short novels The The concept of number is one of the oldest and most Third Man and The Tenth Man; 5 and 9 by Dorothy useful concepts in the history of human race.2–5 People Leigh Sayers (1893–1957) in the titles of her nov- have been aware of the number concept from the very els The Five Red Herrings and The Nine Taylors.
    [Show full text]
  • Conjecture of Twin Primes (Still Unsolved Problem in Number Theory) an Expository Essay
    Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 12 (2017), 229 { 252 CONJECTURE OF TWIN PRIMES (STILL UNSOLVED PROBLEM IN NUMBER THEORY) AN EXPOSITORY ESSAY Hayat Rezgui Abstract. The purpose of this paper is to gather as much results of advances, recent and previous works as possible concerning the oldest outstanding still unsolved problem in Number Theory (and the most elusive open problem in prime numbers) called "Twin primes conjecture" (8th problem of David Hilbert, stated in 1900) which has eluded many gifted mathematicians. This conjecture has been circulating for decades, even with the progress of contemporary technology that puts the whole world within our reach. So, simple to state, yet so hard to prove. Basic Concepts, many and varied topics regarding the Twin prime conjecture will be cover. Petronas towers (Twin towers) Kuala Lumpur, Malaysia 2010 Mathematics Subject Classification: 11A41; 97Fxx; 11Yxx. Keywords: Twin primes; Brun's constant; Zhang's discovery; Polymath project. ****************************************************************************** http://www.utgjiu.ro/math/sma 230 H. Rezgui Contents 1 Introduction 230 2 History and some interesting deep results 231 2.1 Yitang Zhang's discovery (April 17, 2013)............... 236 2.2 "Polymath project"........................... 236 2.2.1 Computational successes (June 4, July 27, 2013)....... 237 2.2.2 Spectacular progress (November 19, 2013)........... 237 3 Some of largest (titanic & gigantic) known twin primes 238 4 Properties 240 5 First twin primes less than 3002 241 6 Rarefaction of twin prime numbers 244 7 Conclusion 246 1 Introduction The prime numbers's study is the foundation and basic part of the oldest branches of mathematics so called "Arithmetic" which supposes the establishment of theorems.
    [Show full text]
  • Code Library
    Code Library Himemiya Nanao @ Perfect Freeze September 13, 2013 Contents 1 Data Structure 1 1.1 atlantis .......................................... 1 1.2 binary indexed tree ................................... 3 1.3 COT ............................................ 4 1.4 hose ........................................... 7 1.5 Leist tree ........................................ 8 1.6 Network ......................................... 10 1.7 OTOCI ........................................... 16 1.8 picture .......................................... 19 1.9 Size Blanced Tree .................................... 22 1.10 sparse table - rectangle ................................. 27 1.11 sparse table - square ................................... 28 1.12 sparse table ....................................... 29 1.13 treap ........................................... 29 2 Geometry 32 2.1 3D ............................................. 32 2.2 3DCH ........................................... 36 2.3 circle's area ....................................... 40 2.4 circle ........................................... 44 2.5 closest point pair .................................... 45 2.6 half-plane intersection ................................. 49 2.7 intersection of circle and poly .............................. 52 2.8 k-d tree .......................................... 53 2.9 Manhattan MST ..................................... 56 2.10 rotating caliper ...................................... 60 2.11 shit ............................................ 63 2.12 other ..........................................
    [Show full text]
  • Statistics of Random Permutations and the Cryptanalysis of Periodic Block Ciphers
    c de Gruyter 2008 J. Math. Crypt. 2 (2008), 1–20 DOI 10.1515 / JMC.2008.xxx Statistics of Random Permutations and the Cryptanalysis Of Periodic Block Ciphers Nicolas T. Courtois, Gregory V. Bard, and Shaun V. Ault Communicated by xxx Abstract. A block cipher is intended to be computationally indistinguishable from a random permu- tation of appropriate domain and range. But what are the properties of a random permutation? By the aid of exponential and ordinary generating functions, we derive a series of collolaries of interest to the cryptographic community. These follow from the Strong Cycle Structure Theorem of permu- tations, and are useful in rendering rigorous two attacks on Keeloq, a block cipher in wide-spread use. These attacks formerly had heuristic approximations of their probability of success. Moreover, we delineate an attack against the (roughly) millionth-fold iteration of a random per- mutation. In particular, we create a distinguishing attack, whereby the iteration of a cipher a number of times equal to a particularly chosen highly-composite number is breakable, but merely one fewer round is considerably more secure. We then extend this to a key-recovery attack in a “Triple-DES” style construction, but using AES-256 and iterating the middle cipher (roughly) a million-fold. It is hoped that these results will showcase the utility of exponential and ordinary generating functions and will encourage their use in cryptanalytic research. Keywords. Generating Functions, EGF, OGF, Random Permutations, Cycle Structure, Cryptanaly- sis, Iterations of Permutations, Analytic Combinatorics, Keeloq. AMS classification. 05A15, 94A60, 20B35, 11T71. 1 Introduction The technique of using a function of a variable to count objects of various sizes, using the properties of multiplication and addition of series as an aid, is accredited to Pierre- Simon Laplace [12].
    [Show full text]