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A Clasification of Known Root Prime-Generating
Special properties of the first absolute Fermat pseudoprime, the number 561 Marius Coman Bucuresti, Romania email: [email protected] Abstract. Though is the first Carmichael number, the number 561 doesn’t have the same fame as the third absolute Fermat pseudoprime, the Hardy-Ramanujan number, 1729. I try here to repair this injustice showing few special properties of the number 561. I will just list (not in the order that I value them, because there is not such an order, I value them all equally as a result of my more or less inspired work, though they may or not “open a path”) the interesting properties that I found regarding the number 561, in relation with other Carmichael numbers, other Fermat pseudoprimes to base 2, with primes or other integers. 1. The number 2*(3 + 1)*(11 + 1)*(17 + 1) + 1, where 3, 11 and 17 are the prime factors of the number 561, is equal to 1729. On the other side, the number 2*lcm((7 + 1),(13 + 1),(19 + 1)) + 1, where 7, 13 and 19 are the prime factors of the number 1729, is equal to 561. We have so a function on the prime factors of 561 from which we obtain 1729 and a function on the prime factors of 1729 from which we obtain 561. Note: The formula N = 2*(d1 + 1)*...*(dn + 1) + 1, where d1, d2, ...,dn are the prime divisors of a Carmichael number, leads to interesting results (see the sequence A216646 in OEIS); the formula M = 2*lcm((d1 + 1),...,(dn + 1)) + 1 also leads to interesting results (see the sequence A216404 in OEIS). -
Generalizing Benford's Law Using Power Laws
Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 970284, 10 pages doi:10.1155/2009/970284 Research Article Generalizing Benford’s Law Using Power Laws: Application to Integer Sequences Werner Hurlimann¨ Feldstrasse 145, CH-8004 Zurich,¨ Switzerland Correspondence should be addressed to Werner Hurlimann,¨ [email protected] Received 25 March 2009; Revised 16 July 2009; Accepted 19 July 2009 Recommended by Kenneth Berenhaut Many distributions for first digits of integer sequences are not Benford. A simple method to derive parametric analytical extensions of Benford’s law for first digits of numerical data is proposed. Two generalized Benford distributions are considered, namely, the two-sided power Benford TSPB distribution, which has been introduced in Hurlimann¨ 2003, and the new Pareto Benford PB distribution. Based on the minimum chi-square estimators, the fitting capabilities of these generalized Benford distributions are illustrated and compared at some interesting and important integer sequences. In particular, it is significant that much of the analyzed integer sequences follow with a high P-value the generalized Benford distributions. While the sequences of prime numbers less than 1000, respectively, 10 000 are not at all Benford or TSPB distributed, they are approximately PB distributed with high P-values of 93.3% and 99.9% and reveal after a further deeper analysis of longer sequences a new interesting property. On the other side, Benford’s law of a mixing of data sets is rejected at the 5% significance level while the PB law is accepted with a 93.6% P-value, which improves the P-value of 25.2%, which has been obtained previously for the TSPB law. -
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 -
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. -
Number Theory Boring? Nein!
Running head: PATTERNS IN NINE 1 Number Theory Boring? Nein! Discerning Patterns in Multiples of Nine Gina Higgins Mathematical Evolutions Jenny McCarthy, Jonathan Phillips Summer Ventures in Science and Mathematics The University of North Carolina at Charlotte PATTERNS IN NINE 2 Abstract This is the process Gina Higgins went through in order to prove that the digital sum of a whole, natural number multiplied by nine would always equal nine. This paper gives a brief history on the number nine and gives a color coded, three-hundred row chart of multiplies of nine as an example and reference. Basic number theory principles were applied with Mathematical Implementation to successfully create a method that confirmed the hypothesis that the digital root of every product of nine equals nine. From this method could evolve into possible future attempts to prove why the digital root anomaly occurred. PATTERNS IN NINE 3 The number nine is the largest digit in a based ten number (decimal) system. In many ancient cultures such as China, India and Egypt, the number nine was a symbol of strength representing royalty, religion or powerful enemies (Mackenzie, 2005). The symbol for the number nine used most commonly today developed from the Hindu-Arabic System written in the third century. Those symbols developed from the Brahmi numerals found as far back as 300 BC. (Ifrah, 1981) Figure 1 Number theory is a branch of mathematics dealing specifically with patterns and sequences found in whole numbers. It was long considered the purest form of mathematics because there was not a practical application until the 20th century when the computer was invented. -
INDEX to ANSWERS Issue Page Number Symmetryplus 60 2
INDEX TO ANSWERS Issue Page number SYMmetryplus 60 2 SYMmetryplus 61 4 SYMmetryplus 62 5 SYMmetryplus 63 8 SYMmetryplus 64 9 SYMmetryplus 65 10 SYMmetryplus 66 13 SYMmetryplus 67 15 SYMmetryplus 68 17 SYMmetryplus 69 21 SYMmetryplus 70 25 SYMmetryplus 71 29 SYMmetryplus 72 33 SYMmetryplus 73 37 SYMmetryplus 74 44 1 ANSWERS FROM ISSUE 60 SOME TRIANGLE NUMBERS – 2 Many thanks to Andrew Palfreyman who found five, not four solutions! 7 7 7 7 7 1 0 5 3 0 0 4 0 6 9 0 3 9 4 6 3 3 3 3 1 Grid A 6 6 1 3 2 6 2 0 1 6 0 0 Grid B CROSSNUMBER Many thanks again to Andrew Palfreyman who pointed out that 1 Down and 8 Across do not give unique answers so there are four possible solutions. 1 2 1 2 1 2 1 2 1 4 4 8 1 4 4 8 1 4 4 8 1 4 4 8 3 3 3 3 3 9 1 9 8 9 1 9 3 9 1 9 8 9 1 9 4 5 4 5 4 5 4 5 2 3 1 0 2 3 1 0 2 3 1 0 2 3 1 0 6 7 6 7 6 7 6 7 1 0 9 8 1 0 9 8 1 0 9 8 1 0 9 8 8 9 8 9 8 9 8 9 3 6 1 0 3 6 1 0 9 6 1 0 9 6 1 0 10 10 10 10 5 3 4 3 5 3 4 3 5 3 4 3 5 3 4 3 TREASURE HUNTS 12, 13 12 This is a rostral column in St Petersburg, Russia. -
Some Results on Benford's Law and Ulam Sequences
Some Results on Benford's Law and Ulam Sequences Nicholas Alvarez,´ Andrew Hwang, Aaron Kriegman Mentor: Jayadev Athreya September 28, 2017 Abstract In this paper, we give an overview on the ubiquitous nature of Benford's Law and cover principles of equidistribution to answer questions concerning under which conditions Benford's Law is satisfied. The paper expands to cover Benford's law for different bases, exponential sequences, recursive sequences, regular sequences, and certain Ulam sequences. By gaining a deeper insight of the criteria for sequences to satisfy Beford's Law, we were able to draw out conclusions about the possible linear growth of certain Ulam Sequences, particularly the standard (1,2) Ulam sequence. Furthermore, we establish an undiscovered, greater structure found within Ulam sequences, which sheds insight on the possibility of a formula to calculate all Ulam numbers. 1 Introduction The mysterious phenomena that characterizes the essence of Benfords Law lies in an observation of the frequency distribution of the leading digits in both very abstract and many real-life sets of numerical data (the leading digit of a number is its leftmost digit). It is natural to seek the criteria a set or sequence must satisfy in order to obey this intriguing frequency distribution. Many distinctly defined sets and sequences satisfy Benfords Law, and our group was able to find infinitely many more examples of sequences that satisfy this law. As we discover more of these criteria, we begin to uncover structures in seemingly unnatural sets. By running extensive code, we can take sets of numerical data and calculate the frequency of leading digits and observe whether these sets satisfy Benfords Law for an arbitrarily large number of terms. -
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 -
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. -
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. -
Digital Roots and Their Properties
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. III (Jan - Feb. 2015), PP 19-24 www.iosrjournals.org Digital Roots and Their Properties Aprajita1, Amar Kumar2 1(Department of Mathematics, J R S College, Jamalpur T M Bhagalpur University Bhagalpur, INDIA) 2(Department of Mathematics, J R S College, Jamalpur T M Bhagalpur University Bhagalpur, INDIA) Abstract: In this paper we want to highlight the properties of digital root and some result based on digital root. There are many interesting properties of digital root. We can also study nature of factors of a given number based on Vedic square and then we can implement this concept for an algorithm to find out a number is prime or not. Keywords: Digital Roots, factors, Integral roots, partition, Vedic square, Primality test. I. Introduction In recreational number theory (A. Gray [1]]) was the first to apply “digital root” but their application had not been previously recognized as relevant. Many other authors have also discussed on the same topics on different way but here we try to discuss the same in a manner which is different from all. Generally, digital root of any natural number is the digit obtained by adding the digits of the number till a single digit is obtained (see [2]) . We can define digital root of a number as: dr: N D where N is the set of natural numbers and D 1,2,3,4,5,6,7,8,9 and dr() a sum of digits of the number „a‟ till single digit is obtained. -
The Ulam Numbers up to One Trillion
The Ulam Numbers up to One Trillion Philip Gibbs and Judson McCranie All Ulam numbers up to one trillion are computed using an efficient linear-time algorithm. We report on the distribution of the numbers including the positions of the largest gaps and we provide an explanation for the previously observed period of clustering. Introduction The Ulam numbers were introduced in 1964 by Stanislaw Ulam as a sequence of integers with a simple additive definition that provide an apparently pseudo-random behaviour [1]. The sequence of Ulam numbers begins with the two integers 푈1 = 1 and 푈2 = 2. Each subsequent Ulam number 푈푛 is the smallest integer that is the sum of two distinct smaller Ulam numbers in exactly one way. It continues 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, … In this study we have calculated all Ulam numbers less than one trillion. There are 73,979,274,540 Ulam numbers up to this limit. The Ulam numbers form an infinite sequence. In the worst case an Ulam number 푈푛 is the sum of 푈푛−1 and 푈푛−3 Therefore 푈푛 ≤ 푁푛−2 where {푁푛} are the numbers in Narayana’s cows sequence: 1,1,1,2,3,4,6,9,13,19,… with the recurrence relation 푁푛 = 푁푛−1 + 푁푛−3 [2]. A lower bound can be set by observing that given five consecutive numbers greater than five, at most two can be Ulam 푈 numbers so 푈 ≥ 2.5푛 − 7. These limits imply that the asymptotic density 퐷 = lim 푛 of Ulam 푛 푛→∞ 푛 numbers in the positive integers is constrained by 0 ≤ 퐷 ≤ 0.4.