Properties of Prime Numbers
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Mersenne and Fermat Numbers 2, 3, 5, 7, 13, 17, 19, 31
MERSENNE AND FERMAT NUMBERS RAPHAEL M. ROBINSON 1. Mersenne numbers. The Mersenne numbers are of the form 2n — 1. As a result of the computation described below, it can now be stated that the first seventeen primes of this form correspond to the following values of ra: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281. The first seventeen even perfect numbers are therefore obtained by substituting these values of ra in the expression 2n_1(2n —1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 2127—1, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in Archibald [l]; see also Kraitchik [4]. The next five Mersenne primes were found in 1952; they are at present the five largest known primes of any form. They were announced in Lehmer [7] and discussed by Uhler [13]. It is clear that 2" —1 can be factored algebraically if ra is composite; hence 2n —1 cannot be prime unless w is prime. Fermat's theorem yields a factor of 2n —1 only when ra + 1 is prime, and hence does not determine any additional cases in which 2"-1 is known to be com- posite. On the other hand, it follows from Euler's criterion that if ra = 0, 3 (mod 4) and 2ra + l is prime, then 2ra + l is a factor of 2n— 1. -
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. -
Primes and Primality Testing
Primes and Primality Testing A Technological/Historical Perspective Jennifer Ellis Department of Mathematics and Computer Science What is a prime number? A number p greater than one is prime if and only if the only divisors of p are 1 and p. Examples: 2, 3, 5, and 7 A few larger examples: 71887 524287 65537 2127 1 Primality Testing: Origins Eratosthenes: Developed “sieve” method 276-194 B.C. Nicknamed Beta – “second place” in many different academic disciplines Also made contributions to www-history.mcs.st- geometry, approximation of andrews.ac.uk/PictDisplay/Eratosthenes.html the Earth’s circumference Sieve of Eratosthenes 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Sieve of Eratosthenes We only need to “sieve” the multiples of numbers less than 10. Why? (10)(10)=100 (p)(q)<=100 Consider pq where p>10. Then for pq <=100, q must be less than 10. By sieving all the multiples of numbers less than 10 (here, multiples of q), we have removed all composite numbers less than 100. -
Prime Numbers and Cryptography
the sunday tImes of malta maRCh 11, 2018 | 51 LIFE &WELLBEING SCIENCE Prime numbers and cryptography ALEXANDER FARRUGIA factored into its two primes by Benjamin Moody – that is a number having 155 digits! (By comparison, the number You are given a number n, and you 6,436,609 has only seven digits.) The first 50 prime numbers. are asked to find the two whole It took his desktop computer 73 numbers a and b, both greater days to achieve this feat. Later, than 1, such that a times b is equal during the same year, a 768-bit MYTH to n. If I give you the number 21, number, having 232 digits, was say, then you would tell me “that factored into its constituent DEBUNKED is 7 times 3”. Or, if I give you 55, primes by a team of 13 academ - you would say “that is 11 times 5”. ics – it took them two years to do Now suppose I ask you for the so! This is the largest number Is 1 a prime two whole numbers, both greater that has been factored to date. than 1, whose product (multipli - Nowadays, most websites number? cation) equals the number employing the RSA algorithm If this question was asked 6,436,609. This is a much harder use at least a 2,048-bit (617- before the start of the 20th problem, right? I invite you to try digit) number to encrypt data. century, one would have to work it out, before you read on. Some websites such as Facebook invariably received a ‘yes’. -
Arxiv:1412.5226V1 [Math.NT] 16 Dec 2014 Hoe 11
q-PSEUDOPRIMALITY: A NATURAL GENERALIZATION OF STRONG PSEUDOPRIMALITY JOHN H. CASTILLO, GILBERTO GARC´IA-PULGAR´IN, AND JUAN MIGUEL VELASQUEZ-SOTO´ Abstract. In this work we present a natural generalization of strong pseudoprime to base b, which we have called q-pseudoprime to base b. It allows us to present another way to define a Midy’s number to base b (overpseudoprime to base b). Besides, we count the bases b such that N is a q-probable prime base b and those ones such that N is a Midy’s number to base b. Furthemore, we prove that there is not a concept analogous to Carmichael numbers to q-probable prime to base b as with the concept of strong pseudoprimes to base b. 1. Introduction Recently, Grau et al. [7] gave a generalization of Pocklignton’s Theorem (also known as Proth’s Theorem) and Miller-Rabin primality test, it takes as reference some works of Berrizbeitia, [1, 2], where it is presented an extension to the concept of strong pseudoprime, called ω-primes. As Grau et al. said it is right, but its application is not too good because it is needed m-th primitive roots of unity, see [7, 12]. In [7], it is defined when an integer N is a p-strong probable prime base a, for p a prime divisor of N −1 and gcd(a, N) = 1. In a reading of that paper, we discovered that if a number N is a p-strong probable prime to base 2 for each p prime divisor of N − 1, it is actually a Midy’s number or a overpseu- doprime number to base 2. -
WXML Final Report: Prime Spacings
WXML Final Report: Prime Spacings Jayadev Athreya, Barry Forman, Kristin DeVleming, Astrid Berge, Sara Ahmed Winter 2017 1 Introduction 1.1 The initial problem In mathematics, prime gaps (defined as r = pn − pn−1) are widely studied due to the potential insights they may reveal about the distribution of the primes, as well as methods they could reveal about obtaining large prime numbers. Our research of prime spacings focused on studying prime intervals, which th is defined as pn − pn−1 − 1, where pn is the n prime. Specifically, we looked at what we defined as prime-prime intervals, that is primes having pn − pn−1 − 1 = r where r is prime. We have also researched how often different values of r show up as prime gaps or prime intervals, and why this distribution of values of r occurs the way that we have experimentally observed it to. 1.2 New directions The idea of studying the primality of prime intervals had not been previously researched, so there were many directions to take from the initial proposed problem. We saw multiple areas that needed to be studied: most notably, the prevalence of prime-prime intervals in the set of all prime intervals, and the distribution of the values of prime intervals. 1 2 Progress 2.1 Computational The first step taken was to see how many prime intervals up to the first nth prime were prime. Our original thoughts were that the amount of prime- prime intervals would grow at a rate of π(π(x)), since the function π tells us how many primes there are up to some number x. -
Appendix a Tables of Fermat Numbers and Their Prime Factors
Appendix A Tables of Fermat Numbers and Their Prime Factors The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. Carl Friedrich Gauss Disquisitiones arithmeticae, Sec. 329 Fermat Numbers Fo =3, FI =5, F2 =17, F3 =257, F4 =65537, F5 =4294967297, F6 =18446744073709551617, F7 =340282366920938463463374607431768211457, Fs =115792089237316195423570985008687907853 269984665640564039457584007913129639937, Fg =134078079299425970995740249982058461274 793658205923933777235614437217640300735 469768018742981669034276900318581864860 50853753882811946569946433649006084097, FlO =179769313486231590772930519078902473361 797697894230657273430081157732675805500 963132708477322407536021120113879871393 357658789768814416622492847430639474124 377767893424865485276302219601246094119 453082952085005768838150682342462881473 913110540827237163350510684586298239947 245938479716304835356329624224137217. The only known Fermat primes are Fo, ... , F4 • 208 17 lectures on Fermat numbers Completely Factored Composite Fermat Numbers m prime factor year discoverer 5 641 1732 Euler 5 6700417 1732 Euler 6 274177 1855 Clausen 6 67280421310721* 1855 Clausen 7 59649589127497217 1970 Morrison, Brillhart 7 5704689200685129054721 1970 Morrison, Brillhart 8 1238926361552897 1980 Brent, Pollard 8 p**62 1980 Brent, Pollard 9 2424833 1903 Western 9 P49 1990 Lenstra, Lenstra, Jr., Manasse, Pollard 9 p***99 1990 Lenstra, Lenstra, Jr., Manasse, Pollard -
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 -
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. -
The Aliquot Constant We first Show the Following Result on the Geometric Mean for the Ordi- Nary Sum of Divisors Function
THE ALIQUOT CONSTANT WIEB BOSMA AND BEN KANE Abstract. The average value of log s(n)/n taken over the first N even integers is shown to converge to a constant λ when N tends to infinity; moreover, the value of this constant is approximated and proven to be less than 0. Here s(n) sums the divisors of n less than n. Thus the geometric mean of s(n)/n, the growth factor of the function s, in the long run tends to be less than 1. This could be interpreted as probabilistic evidence that aliquot sequences tend to remain bounded. 1. Introduction This paper is concerned with the average growth of aliquot sequences. An aliquot sequence is a sequence a0, a1, a2,... of positive integers ob- tained by iteration of the sum-of-aliquot-divisors function s, which is defined for n> 1 by s(n)= d. d|n Xd<n The aliquot sequence with starting value a0 is then equal to 2 a0, a1 = s(a0), a2 = s(a1)= s (a0),... ; k we will say that the sequence terminates (at 1) if ak = s (a0)=1 for some k ≥ 0. The sequence cycles (or is said to end in a cycle) k l 0 if s (a0) = s (a0) for some k,l with 0 ≤ l<k, where s (n) = n by arXiv:0912.3660v1 [math.NT] 18 Dec 2009 definition. Note that s is related to the ordinary sum-of-divisors function σ, with σ(n)= d|n d, by s(n)= σ(n) − n for integers n> 1. -
A Clasification of Known Root Prime-Generating
Two exciting classes of odd composites defined by a relation between their prime factors Marius Coman Bucuresti, Romania email: [email protected] Abstract. In this paper I will define two interesting classes of odd composites often met (by the author of this paper) in the study of Fermat pseudoprimes, which might also have applications in the study of big semiprimes or in other fields. This two classes of composites n = p(1)*...*p(k), where p(1), ..., p(k) are the prime factors of n are defined in the following way: p(j) – p(i) + 1 is a prime or a power of a prime, respectively p(i) + p(j) – 1 is a prime or a power of prime for any p(i), p(j) prime factors of n such that p(1) ≤ p(i) < p(j) ≤ p(k). Definition 1: We name the odd composites n = p(1)*...*p(k), where p(1), ..., p(k) are the prime factors of n, with the property that p(j) – p(i) + 1 is a prime or a power of a prime for any p(i), p(j) prime factors of n such that p(1) ≤ p(i) < p(j) ≤ p(k), Coman composites of the first kind. If n = p*q is a squarefree semiprime, p < q, with the property that q – p + 1 is a prime or a power of a prime, then n it will be a Coman semiprime of the first kind. Examples: : 2047 = 23*89 is a Coman semiprime of the first kind because 89 – 23 + 1 = 67, a prime; : 4681 = 31*151 is a Coman semiprime of the first kind because 151 – 31 + 1 = 121, a power of a prime; : 1729 = 7*13*19 is a Coman composite of the first kind because 19 – 7 + 1 = 13, a prime, 19 – 13 + 1 = 7, a prime, and 13 – 7 + 1 = 7, a prime.