Number Theory for Mathematical Contests
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08 Cyclotomic Polynomials
8. Cyclotomic polynomials 8.1 Multiple factors in polynomials 8.2 Cyclotomic polynomials 8.3 Examples 8.4 Finite subgroups of fields 8.5 Infinitude of primes p = 1 mod n 8.6 Worked examples 1. Multiple factors in polynomials There is a simple device to detect repeated occurrence of a factor in a polynomial with coefficients in a field. Let k be a field. For a polynomial n f(x) = cnx + ::: + c1x + c0 [1] with coefficients ci in k, define the (algebraic) derivative Df(x) of f(x) by n−1 n−2 2 Df(x) = ncnx + (n − 1)cn−1x + ::: + 3c3x + 2c2x + c1 Better said, D is by definition a k-linear map D : k[x] −! k[x] defined on the k-basis fxng by D(xn) = nxn−1 [1.0.1] Lemma: For f; g in k[x], D(fg) = Df · g + f · Dg [1] Just as in the calculus of polynomials and rational functions one is able to evaluate all limits algebraically, one can readily prove (without reference to any limit-taking processes) that the notion of derivative given by this formula has the usual properties. 115 116 Cyclotomic polynomials [1.0.2] Remark: Any k-linear map T of a k-algebra R to itself, with the property that T (rs) = T (r) · s + r · T (s) is a k-linear derivation on R. Proof: Granting the k-linearity of T , to prove the derivation property of D is suffices to consider basis elements xm, xn of k[x]. On one hand, D(xm · xn) = Dxm+n = (m + n)xm+n−1 On the other hand, Df · g + f · Dg = mxm−1 · xn + xm · nxn−1 = (m + n)xm+n−1 yielding the product rule for monomials. -
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. -
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. -
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. -
Solutions and Investigations
UKMT UKMT UKMT Junior Mathematical Challenge Thursday 30th April 2015 Organised by the United Kingdom Mathematics Trust supported by Solutions and investigations These solutions augment the printed solutions that we send to schools. For convenience, the solutions sent to schools are confined to two sides of A4 paper and therefore in many casesare rather short. The solutions given here have been extended. In some cases we give alternative solutions, and we have included some exercises for further investigation. We welcome comments on these solutions. Please send them to [email protected]. The Junior Mathematical Challenge (JMC) is a multiple-choice paper. For each question, you are presented with five options, of which just one is correct. It follows that often you canfindthe correct answers by working backwards from the given alternatives, or by showing that four of them are not correct. This can be a sensible thing to do in the context of the JMC. However, this does not provide a full mathematical explanation that would be acceptable if you were just given the question without any alternative answers. So for each question we have included a complete solution which does not use the fact that one of the given alternatives is correct. Thus we have aimed to give full solutions with all steps explained. We therefore hope that these solutions can be used as a model for the type of written solution that is expected when presenting a complete solution to a mathematical problem (for example, in the Junior Mathematical Olympiad and similar competitions). These solutions may be used freely within your school or college. -
Sequences of Numbers Obtained by Digit and Iterative Digit Sums of Sophie Germain Primes and Its Variants
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 2 (2016), pp. 1473-1480 © Research India Publications http://www.ripublication.com Sequences of numbers obtained by digit and iterative digit sums of Sophie Germain primes and its variants 1Sheila A. Bishop, *1Hilary I. Okagbue, 2Muminu O. Adamu and 3Funminiyi A. Olajide 1Department of Mathematics, Covenant University, Canaanland, Ota, Nigeria. 2Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria. 3Department of Computer and Information Sciences, Covenant University, Canaanland, Ota, Nigeria. Abstract Sequences were generated by the digit and iterative digit sums of Sophie Germain and Safe primes and their variants. The results of the digit and iterative digit sum of Sophie Germain and Safe primes were almost the same. The same applied to the square and cube of the respective primes. Also, the results of the digit and iterative digit sum of primes that are not Sophie Germain are the same with the primes that are notSafe. The results of the digit and iterative digit sum of prime that are either Sophie Germain or Safe are like the combination of the results of the respective primes when considered separately. Keywords: Sophie Germain prime, Safe prime, digit sum, iterative digit sum. Introduction In number theory, many types of primes exist; one of such is the Sophie Germain prime. A prime number p is said to be a Sophie Germain prime if 21p is also prime. A prime number qp21 is known as a safe prime. Sophie Germain prime were named after the famous French mathematician, physicist and philosopher Sophie Germain (1776-1831). -
Arxiv:2106.08994V2 [Math.GM] 1 Aug 2021 Efc Ubr N30b.H Rvdta F2 If That and Proved Properties He Studied BC
Measuring Abundance with Abundancy Index Kalpok Guha∗ Presidency University, Kolkata Sourangshu Ghosh† Indian Institute of Technology Kharagpur, India Abstract A positive integer n is called perfect if σ(n) = 2n, where σ(n) denote n σ(n) the sum of divisors of . In this paper we study the ratio n . We de- I → I n σ(n) fine the function Abundancy Index : N Q with ( ) = n . Then we study different properties of Abundancy Index and discuss the set of Abundancy Index. Using this function we define a new class of num- bers known as superabundant numbers. Finally we study superabundant numbers and their connection with Riemann Hypothesis. 1 Introduction Definition 1.1. A positive integer n is called perfect if σ(n)=2n, where σ(n) denote the sum of divisors of n. The first few perfect numbers are 6, 28, 496, 8128, ... (OEIS A000396), This is a well studied topic in number theory. Euclid studied properties and nature of perfect numbers in 300 BC. He proved that if 2p −1 is a prime, then 2p−1(2p −1) is an even perfect number(Elements, Prop. IX.36). Later mathematicians have arXiv:2106.08994v2 [math.GM] 1 Aug 2021 spent years to study the properties of perfect numbers. But still many questions about perfect numbers remain unsolved. Two famous conjectures related to perfect numbers are 1. There exist infinitely many perfect numbers. Euler [1] proved that a num- ber is an even perfect numbers iff it can be written as 2p−1(2p − 1) and 2p − 1 is also a prime number. -
Problems Archives
Cache update: 56 minutes Problems Archives The problems archives table shows problems 1 to 651. If you would like to tackle the 10 most recently published problems then go to Recent problems. Click the description/title of the problem to view details and submit your answer. ID Description / Title Solved By 1 Multiples of 3 and 5 834047 2 Even Fibonacci numbers 666765 3 Largest prime factor 476263 4 Largest palindrome product 422115 5 Smallest multiple 428955 6 Sum square difference 431629 7 10001st prime 368958 8 Largest product in a series 309633 9 Special Pythagorean triplet 313806 10 Summation of primes 287277 11 Largest product in a grid 207029 12 Highly divisible triangular number 194069 13 Large sum 199504 14 Longest Collatz sequence 199344 15 Lattice paths 164576 16 Power digit sum 201444 17 Number letter counts 133434 18 Maximum path sum I 127951 19 Counting Sundays 119066 20 Factorial digit sum 175533 21 Amicable numbers 128681 22 Names scores 118542 23 Non-abundant sums 91300 24 Lexicographic permutations 101261 25 1000-digit Fibonacci number 137312 26 Reciprocal cycles 73631 27 Quadratic primes 76722 28 Number spiral diagonals 96208 29 Distinct powers 92388 30 Digit fifth powers 96765 31 Coin sums 74310 32 Pandigital products 62296 33 Digit cancelling fractions 62955 34 Digit factorials 82985 35 Circular primes 74645 36 Double-base palindromes 78643 37 Truncatable primes 64627 38 Pandigital multiples 55119 39 Integer right triangles 64132 40 Champernowne's constant 70528 41 Pandigital prime 59723 42 Coded triangle numbers 65704 43 Sub-string divisibility 52160 44 Pentagon numbers 50757 45 Triangular, pentagonal, and hexagonal 62652 46 Goldbach's other conjecture 53607 47 Distinct primes factors 50539 48 Self powers 100136 49 Prime permutations 50577 50 Consecutive prime sum 54478 Cache update: 56 minutes Problems Archives The problems archives table shows problems 1 to 651. -
From Vortex Mathematics to Smith Numbers: Demystifying Number Structures and Establishing Sieves Using Digital Root
SCITECH Volume 15, Issue 3 RESEARCH ORGANISATION Published online: November 21, 2019| Journal of Progressive Research in Mathematics www.scitecresearch.com/journals From Vortex Mathematics to Smith Numbers: Demystifying Number Structures and Establishing Sieves Using Digital Root Iman C Chahine1 and Ahmad Morowah2 1College of Education, University of Massachusetts Lowell, MA 01854, USA 2Retired Mathematics Teacher, Beirut, Lebanon Corresponding author email: [email protected] Abstract: Proficiency in number structures depends on a continuous development and blending of intricate combinations of different types of numbers and its related characteristics. The purpose of this paper is to unpack the mechanisms and underlying notions that elucidate the potential process of number construction and its inherent structures. By employing the concept of digital root, we show how juxtaposed assumptions can play in delineating generalized models of number structures bridging the abstract, the numerical, and the physical worlds. While there are numerous proposed ways of constructing Smith numbers, developing a generalized algorithm could help provide a unified approach to generating number structures with inherent commonalities. In this paper, we devise a sieve for all Smith numbers as well as other related numbers. The sieve works on the principle of digital roots of both Sd(N), the sum of the digits of a number N and that of Sp(N), the sum of the digits of the extended prime divisors of N. Starting with Sp(N) = Sp(p.q.r…), where p,q,r,…, are the prime divisors whose product yields N and whose digital root (n) equals to that of Sd(N) thus Sd(N) = n + 9x; x є N. -
Three Topics in Additive Prime Number Theory 3
THREE TOPICS IN ADDITIVE PRIME NUMBER THEORY BEN GREEN Abstract. We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes. Introduction. These notes are to accompany two lectures I am scheduled to give at the Current Developments in Mathematics conference at Harvard in November 2007. The title of those lectures is ‘A good new millennium for primes’, but I have chosen a rather drier title for these notes for two reasons. Firstly, the title of the lectures was unashamedly stolen (albeit with permission) from Andrew Granville’s entertaining article [16] of the same name. Secondly, and more seriously, I do not wish to claim that the topics chosen here represent a complete survey of developments in prime number theory since 2000 or even a selection of the most important ones. Indeed there are certainly omissions, such as the lack of any discussion of the polynomial-time primality test [2], the failure to even mention the recent work on primes in orbits by Bourgain, Gamburd and Sarnak, and many others. I propose to discuss the following three topics, in greatly varying degrees of depth. Suggestions for further reading will be provided. The three sections may be read inde- pendently although there are links between them. 1. Gaps between primes. -
1.6 Exponents and the Order of Operations
1.6 EXPONENTS AND THE ORDER OF OPERATIONS Writing Whole Numbers and Variables in Exponent Form Student Learning Objectives Recall that in the multiplication problem 3 # 3 # 3 # 3 # 3 = 243 the number 3 is called a factor. We can write the repeated multiplication 3 # 3 # 3 # 3 # 3 using a shorter nota- After studying this section, you will tion, 35, because there are five factors of 3 in the repeated multiplication. We say be able to: 5 5 that 3 is written in exponent form. 3 is read “three to the fifth power.” Write whole numbers and variables in exponent form. Evaluate numerical and EXPONENT FORM algebraic expressions in The small number 5 is called an exponent. Whole number exponents, except exponent form. zero, tell us how many factors are in the repeated multiplication. The number Use symbols and key words 3 is called the base. The base is the number that is multiplied. for expressing exponents. Follow the order of operations. 3 # 3 # 3 # 3 # 3 = 35 The exponent is 5. 3 appears as a factor 5 times. The base is 3. We do not multiply the base 3 by the exponent 5. The 5 just tells us how many 3’s are in the repeated multiplication. If a whole number or variable does not have an exponent visible, the exponent is understood to be 1. 9 = 91 and x = x1 EXAMPLE 1 Write in exponent form. (a)2 # 2 # 2 # 2 # 2 # 2 (b)4 # 4 # 4 # x # x (c) 7 (d) y # y # y # 3 # 3 # 3 # 3 Solution # # # # # = 6 # # # x # x = 3 # x2 3x2 (a)2 2 2 2 2 2 2 (b) 4 4 4 4 or 4 (c)7 = 71 (d) y # y # y # 3 # 3 # 3 # 3 = y3 # 34, or 34y3 Note, it is standard to write the number before the variable in a term. -
Properties of Exponents
1 Algebra II Properties of Exponents 2015-11-09 www.njctl.org 2 Table of Contents Click on topic to go to that section. Review of Integer Exponents Fractional Exponents Exponents with Multiple Terms Identifying Like Terms Evaluating Exponents Using a Calculator Standards 3 Review of Integer Exponents Return to Table of Contents 4 Powers of Integers Just as multiplication is repeated addition, an exponent represents repeated multiplication. For example, a5 read as "a to the fifth power" = a · a · a · a · a In this case, a is the base and 5 is the exponent. The base, a, is multiplied by itself 5 times. 5 Powers of Integers Make sure when you are evaluating exponents of negative numbers, you keep in mind the meaning of the exponent and the rules of multiplication. For example, , which is the same as . However, Notice the difference! Similarly, but . 6 1 Evaluate: 64 Answer 7 2 Evaluate: -128 Answer 8 3 Evaluate: 81 Answer 9 Properties of Exponents The properties of exponents follow directly from expanding them to look at the repeated multiplication they represent. Work to understand the process by which we find these properties and if you can't recall what to do, just repeat these steps to confirm the property. We'll use 3 as the base in our examples, but the properties hold for any base. We show that with base a and powers b and c. We'll use the facts that: 10 Properties of Exponents We need to develop all the properties of exponents so we can discover one of the inverse operations of raising a number to a power which is finding the root of a number.