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An Amazing Prime Heuristic.Pdf
This document has been moved to https://arxiv.org/abs/2103.04483 Please use that version instead. AN AMAZING PRIME HEURISTIC CHRIS K. CALDWELL 1. Introduction The record for the largest known twin prime is constantly changing. For example, in October of 2000, David Underbakke found the record primes: 83475759 264955 1: · The very next day Giovanni La Barbera found the new record primes: 1693965 266443 1: · The fact that the size of these records are close is no coincidence! Before we seek a record like this, we usually try to estimate how long the search might take, and use this information to determine our search parameters. To do this we need to know how common twin primes are. It has been conjectured that the number of twin primes less than or equal to N is asymptotic to N dx 2C2N 2C2 2 2 Z2 (log x) ∼ (log N) where C2, called the twin prime constant, is approximately 0:6601618. Using this we can estimate how many numbers we will need to try before we find a prime. In the case of Underbakke and La Barbera, they were both using the same sieving software (NewPGen1 by Paul Jobling) and the same primality proving software (Proth.exe2 by Yves Gallot) on similar hardware{so of course they choose similar ranges to search. But where does this conjecture come from? In this chapter we will discuss a general method to form conjectures similar to the twin prime conjecture above. We will then apply it to a number of different forms of primes such as Sophie Germain primes, primes in arithmetic progressions, primorial primes and even the Goldbach conjecture. -
Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory
Paul Erd®s and the rise of statistical thinking in elementary number theory Carl Pomerance, Dartmouth College based on the joint survey with Paul Pollack, University of Georgia 1 Let us begin at the beginning: 2 Pythagoras 3 Sum of proper divisors Let s(n) be the sum of the proper divisors of n: For example: s(10) = 1 + 2 + 5 = 8; s(11) = 1; s(12) = 1 + 2 + 3 + 4 + 6 = 16: 4 In modern notation: s(n) = σ(n) − n, where σ(n) is the sum of all of n's natural divisors. The function s(n) was considered by Pythagoras, about 2500 years ago. 5 Pythagoras: noticed that s(6) = 1 + 2 + 3 = 6 (If s(n) = n, we say n is perfect.) And he noticed that s(220) = 284; s(284) = 220: 6 If s(n) = m, s(m) = n, and m 6= n, we say n; m are an amicable pair and that they are amicable numbers. So 220 and 284 are amicable numbers. 7 In 1976, Enrico Bombieri wrote: 8 There are very many old problems in arithmetic whose interest is practically nil, e.g., the existence of odd perfect numbers, problems about the iteration of numerical functions, the existence of innitely many Fermat primes 22n + 1, etc. 9 Sir Fred Hoyle wrote in 1962 that there were two dicult astronomical problems faced by the ancients. One was a good problem, the other was not so good. 10 The good problem: Why do the planets wander through the constellations in the night sky? The not-so-good problem: Why is it that the sun and the moon are the same apparent size? 11 Perfect numbers, amicable numbers, and similar topics were important to the development of elementary number theory. -
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. -
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]). -
On Repdigits As Sums of Fibonacci and Tribonacci Numbers
S S symmetry Article On Repdigits as Sums of Fibonacci and Tribonacci Numbers 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: 17 September 2020; Accepted: 21 October 2020; Published: 26 October 2020 Abstract: In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the easiest case of palindromic numbers, which are a “symmetrical” type of numbers) that can be written in the form Fn + Tn, for some n ≥ 1, where (Fn)n≥0 and (Tn)n≥0 are the sequences of Fibonacci and Tribonacci numbers, respectively. Keywords: Diophantine equations; repdigits; Fibonacci; Tribonacci; Baker’s theory MSC: 11B39; 11J86 1. Introduction A palindromic number is a number that has the same form when written forwards or backwards, i.e., of the form c1c2c3 ... c3c2c1 (thus it can be said that they are “symmetrical” with respect to an axis of symmetry). The first 19th palindromic numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99 and clearly they are a repdigits type. A number n is called repdigit if it has only one repeated digit in its decimal expansion. More precisely, n has the form ! 10` − 1 n = a , 9 for some ` ≥ 1 and a 2 [1, 9] (as usual, we set [a, b] = fa, a + 1, .. -
Sums of Divisors, Perfect Numbers and Factoring*
J. COMPUT. 1986 Society for and Applied Mathematics No. 4, November 1986 02 SUMS OF DIVISORS, PERFECT NUMBERS AND FACTORING* ERIC GARY MILLER5 AND JEFFREY Abstract. Let N be a positive integer, and let denote the sum of the divisors of N = 1+2 +3+6 = 12). We show computing is equivalent to factoring N in the following sense: there is a random polynomial time algorithm that, given N),produces the prime factorization of N, and N) can be computed in polynomial time given the factorization of N. We show that the same result holds for the sum of the kth powers of divisors of We give three new examples of problems that are in Gill’s complexity class BPP perfect numbers, multiply perfect numbers, and amicable pairs. These are the first “natural” sets in BPP that are not obviously in RP. Key words. factoring, sum of divisors, perfect numbers, random reduction, multiply perfect numbers, amicable pairs subject classifications. 1. Introduction. Integer factoring is a well-known difficult problem whose precise computational complexity is still unknown. Several investigators have found algorithms that are much better than the classical method of trial division (see [Guy [Pol], [Dix], [Len]). We are interested in the relationship of factoring to other functions in number theory. It is trivial to show that classical functions like (the number of positive integers less than N and relatively prime to N) can be computed in polynomial time if one can factor N; hence computing is “easier” than factoring. One would also like to find functions “harder” than factoring. -
POWERFUL AMICABLE NUMBERS 1. Introduction Let S(N) := ∑ D Be the Sum of the Proper Divisors of the Natural Number N. Two Disti
POWERFUL AMICABLE NUMBERS PAUL POLLACK P Abstract. Let s(n) := djn; d<n d denote the sum of the proper di- visors of the natural number n. Two distinct positive integers n and m are said to form an amicable pair if s(n) = m and s(m) = n; in this case, both n and m are called amicable numbers. The first example of an amicable pair, known already to the ancients, is f220; 284g. We do not know if there are infinitely many amicable pairs. In the opposite direction, Erd}osshowed in 1955 that the set of amicable numbers has asymptotic density zero. Let ` ≥ 1. A natural number n is said to be `-full (or `-powerful) if p` divides n whenever the prime p divides n. As shown by Erd}osand 1=` Szekeres in 1935, the number of `-full n ≤ x is asymptotically c`x , as x ! 1. Here c` is a positive constant depending on `. We show that for each fixed `, the set of amicable `-full numbers has relative density zero within the set of `-full numbers. 1. Introduction P Let s(n) := djn; d<n d be the sum of the proper divisors of the natural number n. Two distinct natural numbers n and m are said to form an ami- cable pair if s(n) = m and s(m) = n; in this case, both n and m are called amicable numbers. The first amicable pair, 220 and 284, was known already to the Pythagorean school. Despite their long history, we still know very little about such pairs. -
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.