Math in the News: Mersenne Primes
|
|
|
- Jody York
- 5 years ago
- Views:
Transcription
1 Math in the News: Mersenne Primes Helmut Knaust Department of Mathematical Sciences The University of Texas at El Paso El Paso TX Greater El Paso Council of Teachers of Mathematics Fall Conference October 18, 2008
2 1 Math in the News 2 Prime numbers 3 How Many Primes? 4 Mersenne Primes 5 Perfect Numbers 6 Computing Mersenne Primes
3 By the way, the number of atoms in the universe has about 80 digits...
4 Definition and Examples Definition A prime number is a natural number that has exactly two divisors (1 and itself) Here are the first 100 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541
5 Definition and Examples Theorem A natural number p > 1 is prime if and only if no prime number p divides p.
6 Definition and Examples Theorem A natural number p > 1 is prime if and only if no prime number p divides p. For example, to check that 107 is a prime number, we have to check whether 107 is divisible by by 2, 3, 5 and 7.
7 Basic Results and Techniques Another way to find prime numbers is the Sieve of Eratosthenes ( 240 BC):
8 Basic Results and Techniques Another way to find prime numbers is the Sieve of Eratosthenes ( 240 BC):
9 Basic Results and Techniques Another way to find prime numbers is the Sieve of Eratosthenes ( 240 BC):
10 Basic Results and Techniques Another way to find prime numbers is the Sieve of Eratosthenes ( 240 BC):
11 Basic Results and Techniques Another way to find prime numbers is the Sieve of Eratosthenes ( 240 BC):
12 Basic Results and Techniques Another way to find prime numbers is the Sieve of Eratosthenes ( 240 BC):
13 Basic Results and Techniques Another way to find prime numbers is the Sieve of Eratosthenes ( 240 BC):
14 Basic Results and Techniques The next slide shows a graphical representation of the Sieve of Eratosthenes for all numbers = Black dots represent primes, White dots represent composite numbers.
15 Basic Results and Techniques 3
16 Basic Results and Techniques 7
17 Basic Results and Techniques 17
18 Basic Results and Techniques 43
19 Basic Results and Techniques 103
20 Basic Results and Techniques 307
21 Basic Results and Techniques 601
22 Basic Results and Techniques Theorem (Euclid, 300 BC) The Fundamental Theorem of Arithmetic. Every natural number greater than 1 can be written as the product of (one or more) primes. The factorization is unique up to reordering.
23 Basic Results and Techniques Theorem (Euclid, 300 BC) The Fundamental Theorem of Arithmetic. Every natural number greater than 1 can be written as the product of (one or more) primes. The factorization is unique up to reordering. For example: =
24 Euclid s Theorem Theorem (Euclid, 300 BC) There are infinitely many prime numbers.
25 Euclid s Theorem Theorem (Euclid, 300 BC) There are infinitely many prime numbers. Proof: Suppose there are only finitely many primes. More precisely, suppose that {p 1, p 2, p 3,..., p n } is a complete list of all prime numbers.
26 Euclid s Theorem Theorem (Euclid, 300 BC) There are infinitely many prime numbers. Proof: Suppose there are only finitely many primes. More precisely, suppose that {p 1, p 2, p 3,..., p n } is a complete list of all prime numbers. Consider the number N = p 1 p 2 p n + 1.
27 Euclid s Theorem Theorem (Euclid, 300 BC) There are infinitely many prime numbers. Proof: Suppose there are only finitely many primes. More precisely, suppose that {p 1, p 2, p 3,..., p n } is a complete list of all prime numbers. Consider the number N = p 1 p 2 p n + 1. Observe that N is not divisible by any of the primes p 1, p 2, p 3,... p n on our complete list.
28 Euclid s Theorem Now there are two possibilities: If N is prime, we obtain a contradiction since N is not on our complete list of primes.
29 Euclid s Theorem Now there are two possibilities: If N is prime, we obtain a contradiction since N is not on our complete list of primes. If, on the other hand, N is not prime, it must be divisible by one of the primes on our list - but it is not!
30 Euclid s Theorem Now there are two possibilities: If N is prime, we obtain a contradiction since N is not on our complete list of primes. If, on the other hand, N is not prime, it must be divisible by one of the primes on our list - but it is not! In either case we get a contradiction to our assumption that there are only finitely many primes. So there must be infinitely many primes!
31 The Prime Number Theorem The most celebrated result in number theory was conjectured by both Legendre and Gauss at the end of the 18th century, but only proved a century later: Theorem (Hadamard/de la Vallée Poussin, 1896) The Prime Number Theorem. Let Π(n) denote the number of prime numbers n. Then lim n Π(n) n/ ln(n) = 1.
32 The Prime Number Theorem The approximation is pretty slow: n n n ln n n n ln n
33 Classical Conjectures and Recent Results Conjecture (Goldbach, 1742) The Goldbach Conjecture. Every even natural number greater than 2 is the sum of two primes.
34 Classical Conjectures and Recent Results Conjecture (Goldbach, 1742) The Goldbach Conjecture. Every even natural number greater than 2 is the sum of two primes. Here is an example: 46 = (= = = ).
35 Classical Conjectures and Recent Results Conjecture (Goldbach, 1742) The Goldbach Conjecture. Every even natural number greater than 2 is the sum of two primes. Here is an example: 46 = (= = = ). Note that the Goldbach Conjecture would imply that every odd prime number > 5 is the sum of three prime numbers.
36 Classical Conjectures and Recent Results The best known result is Theorem (Vinogradov, 1937) Every large odd prime number is the sum of three primes.
37 Classical Conjectures and Recent Results The best known result is Theorem (Vinogradov, 1937) Every large odd prime number is the sum of three primes. Large currently means greater than
38 Classical Conjectures and Recent Results Conjecture The Twin Prime Conjecture. There are infinitely many pairs of primes with difference 2.
39 Classical Conjectures and Recent Results Conjecture The Twin Prime Conjecture. There are infinitely many pairs of primes with difference 2. Examples are 5 and 7, 461 and 463.
40 Classical Conjectures and Recent Results Conjecture The Twin Prime Conjecture. There are infinitely many pairs of primes with difference 2. Examples are 5 and 7, 461 and 463. The best result known is Theorem (Chen, 1966) There are infinitely many pairs p, p + 2 such that p is prime, and p + 2 is prime or a product of two primes.
41 Classical Conjectures and Recent Results Theorem (Green-Tao Theorem, 2004) There are arbitrarily long sequences of primes in arithmetic progression.
42 Classical Conjectures and Recent Results Theorem (Green-Tao Theorem, 2004) There are arbitrarily long sequences of primes in arithmetic progression. Here is an example of five primes in arithmetic progression: 5, 11, 17, 23, 29. The longest currently known sequence of primes in arithmetic progression was found in It has length 23, starts at , with differences
43 Classical Conjectures and Recent Results Terence Tao, Fields Medal Recipient
44 Definition and Examples Definition A prime number of the form 2 p 1 is called the Mersenne prime M p. The number p is then called a Mersenne exponent.
45 Definition and Examples Definition A prime number of the form 2 p 1 is called the Mersenne prime M p. The number p is then called a Mersenne exponent. Here are some examples: If p = 2, = 3 is a Mersenne prime; p = 7 yields the Mersenne prime = 127.
46 Definition and Examples Definition A prime number of the form 2 p 1 is called the Mersenne prime M p. The number p is then called a Mersenne exponent. Here are some examples: If p = 2, = 3 is a Mersenne prime; p = 7 yields the Mersenne prime = 127. This does not work for all p, even is p is a prime number. For example = 2047 is not a prime: = 23 89
47 Mersenne s Claim Marin Mersenne,
48 Mersenne s Claim Marin Mersenne boldly claimed in 1644 that a complete list of Mersenne exponents less than 258 is given by {2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257}
49 Mersenne s Claim Marin Mersenne boldly claimed in 1644 that a complete list of Mersenne exponents less than 258 is given by {2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257} Here is the list of the corresponding 11 numbers of the form 2 p 1: 3, 7, 31, 127, 8191, , , , , ,
50 Mersenne s Claim Mersenne was wrong two numbers on his list are not primes: = =
51 Mersenne s Claim Mersenne was wrong two numbers on his list are not primes: = = Worse he also missed three exponents: , and are prime numbers.
52 Mersenne s Claim Mersenne s list again: {2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257}
53 Mersenne s Claim Mersenne s list again: {2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257} Here is the corrected list of Mersenne exponents less than 258: {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127}
54 Mersenne s Claim Theorem If a p 1 is a prime number, then a = 2 and p is prime.
55 Mersenne s Claim Theorem If a p 1 is a prime number, then a = 2 and p is prime. Proof: I will only prove that 2 p 1 prime implies that p is prime. Assume that p is not prime, then p = m k, with m, k 2. So 2 p 1 = (2 m ) k 1.
56 Mersenne s Claim Theorem If a p 1 is a prime number, then a = 2 and p is prime. Proof: I will only prove that 2 p 1 prime implies that p is prime. Assume that p is not prime, then p = m k, with m, k 2. So 2 p 1 = (2 m ) k 1. Do you remember how to factor x 3 1? (2 m ) k 1 works the same way:
57 Mersenne s Claim Theorem If a p 1 is a prime number, then a = 2 and p is prime. Proof: I will only prove that 2 p 1 prime implies that p is prime. Assume that p is not prime, then p = m k, with m, k 2. So 2 p 1 = (2 m ) k 1. Do you remember how to factor x 3 1? (2 m ) k 1 works the same way: ( ) (2 m ) k 1 = (2 m 1) (2 m ) k 1 + (2 m ) k 2 + (2 m ) k (2 m ) + 1 End of the story! 2 p 1 is divisible by 2 m 1, and hence not prime.
58 Definition and Examples Definition A natural number is called a perfect number if it equals the sum of its proper divisors.
59 Definition and Examples Definition A natural number is called a perfect number if it equals the sum of its proper divisors. Here are the first two examples: 6 = , 28 =
60 Perfect Numbers and Mersenne Primes As it turns out there is a perfect correspondence between Mersenne primes and even perfect numbers.
61 Perfect Numbers and Mersenne Primes As it turns out there is a perfect correspondence between Mersenne primes and even perfect numbers. Theorem (Euclid, 300 BC) If 2 p 1 is prime, then 2 p 1 (2 p 1) is perfect.
62 Perfect Numbers and Mersenne Primes As it turns out there is a perfect correspondence between Mersenne primes and even perfect numbers. Theorem (Euclid, 300 BC) If 2 p 1 is prime, then 2 p 1 (2 p 1) is perfect. Theorem (Euler, ) If n is an even perfect number, then n = 2 p 1 (2 p 1) for some Mersenne prime 2 p 1.
63 Perfect Numbers and Mersenne Primes As it turns out there is a perfect correspondence between Mersenne primes and even perfect numbers. Theorem (Euclid, 300 BC) If 2 p 1 is prime, then 2 p 1 (2 p 1) is perfect. Theorem (Euler, ) If n is an even perfect number, then n = 2 p 1 (2 p 1) for some Mersenne prime 2 p 1. Conjecture All perfect numbers are even.
64 Known Mersenne Primes # n Digits in Mersenne Number Digits in Perfect number Date of Discovery Discoverer ancient ancient ancient ancient ancient ancient ancient ancient anonymous Cataldi Cataldi Euler Pervushin Powers Powers Lucas January Robinson January Robinson 15 1, June Robinson 16 2, ,327 October Robinson 17 2, ,373 October Robinson 18 3, ,937 September Riesel 19 4,253 1,281 2,561 November Hurwitz 20 4,423 1,332 2,663 November Hurwitz 21 9,689 2,917 5,834 May Gillies 22 9,941 2,993 5,985 May Gillies 23 11,213 3,376 6,751 June Gillies
65 Known Mersenne Primes # n Digits in Mersenne Number Digits in Perfect number Date of Discovery Discoverer 24 19,937 6,002 12,003 March Tuckerman 25 21,701 6,533 13,066 October Noll & Nickel 26 23,209 6,987 13,973 February Noll 27 44,497 13,395 26,790 April Nelson & Slowinski 28 86,243 25,962 51,924 September Slowinski ,503 33,265 66,530 January Colquitt & Welsh ,049 39,751 79,502 September Slowinski ,091 65, ,100 September Slowinski , , ,663 February Slowinski & Gage , , ,430 January Slowinski & Gage 34 1,257, , ,263 September Slowinski & Gage 35 1,398, , ,842 November Armengaud, Woltman et. al. GIMPS 36 2,976, ,932 1,791,864 August Spence, Woltman et. al. GIMPS 37 3,021, ,526 1,819,050 January Clarkson, Woltman, Kurowski et. al. GIMPS & PrimeNet 38 6,972,593 2,098,960 4,197,919 June Hajratwala, Woltman, Kurowski et. al. GIMPS & PrimeNet 39 13,466,917 4,053,946 8,107,892 November Cameron, Woltman, Kurowski et. al. GIMPS & PrimeNet 40* 20,996,011 6,320,430 12,640,858 November Shafer, Woltman, Kurowski et. al. GIMPS & PrimeNet 41* 24,036,583 7,235,733 14,471,465 May Findley, Woltman, Kurowski et. al. GIMPS & PrimeNet 42* 25,964,951 7,816,230 15,632,458 February Nowak, Woltman, Kurowski et. al. GIMPS & PrimeNet 43* 30,402,457 9,152,052 18,304,103 December Cooper, Boone, Woltman, Kurowski et. al. GIMPS & PrimeNet 44* 32,582,657 9,808,358 19,616,714 September Cooper, Boone, Woltman, Kurowski et. al. GIMPS & PrimeNet 45* 37,156,667 11,185,272 22,370,543 September Elvenich, Woltman, Kurowski, et al. GIMPS & PrimeNet 46* 43,112,609 12,978,189 25,957,378 August Smith, Woltman, Kurowski, et al. GIMPS & PrimeNet
66 Lucas-Lehmer Test Nowadays, primality of Mersenne numbers is established with a test developed by Édouard Lucas ( ) and improved by Derrick H. Lehmer ( ): Theorem (Lucas-Lehmer Test, 1870s/1930s) Consider the sequence s 0 = 4, s n = sn for n 1. Let p be an odd prime. Then 2 p 1 is a Mersenne prime if and only if s p 2 = 0 (mod 2 p 1).
67 Lucas-Lehmer Test Here is how this works to check that = 127 is a Mersenne prime: n s n s n (mod 127)
68 Lucas-Lehmer Test Lehmer s Photoelectric Number Sieve
69 GIMPS
70 Conclusion All Questions Answered, All Answers Questioned Contact: Borrowed from Donald Knuth
71 The Lucas - Lehmer Test tu TimeUsed ; z 1; Do n Prime k ; If Nest Function s, Mod s^2 2, 2^n 1, 4, n 2 0, s TimeUsed tu; z ; If s 120, Print z, "\t", n, "\t", IntegerPart s 60, " min.", Print z, "\t", n, "\t", s, " sec." ;,, k, 1, sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec min min min.
72 David Hilbert ( ) We Must Know We Shall Know
The New Largest Known Prime is 2 p 1 With p = Who Cares? Sam Wagstaff Computer Sciences and Mathematics.
The New Largest Known Prime is 2 p 1 With p = 74207281. Who Cares? Sam Wagstaff Computer Sciences and Mathematics November 10, 2016 Earlier in 2016, Cooper, Woltman, Kurowski, Blosser and GIMPS found this
Prime and Perfect Numbers
Prime and Perfect Numbers 0.3 Infinitude of prime numbers 0.3.1 Euclid s proof Euclid IX.20 demonstrates the infinitude of prime numbers. 1 The prime numbers or primes are the numbers 2, 3, 5, 7, 11, 13,
Primes and Factorization
Primes and Factorization 1 A prime number is an integer greater than 1 with no proper divisors. The list begins 2, 3, 5, 7, 11, 13, 19,... See http://primes.utm.edu/ for a wealth of information about primes.
of Nebraska - Lincoln
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership May 2010 Perfect Numbers: Diana French University of
God may not play dice with the universe, but something strange is going on with the prime numbers.
Primes: Definitions God may not play dice with the universe, but something strange is going on with the prime numbers. P. Erdös (attributed by Carl Pomerance) Def: A prime integer is a number whose only
Here is another characterization of prime numbers.
Here is another characterization of prime numbers. Theorem p is prime it has no divisors d that satisfy < d p. Proof [ ] If p is prime then it has no divisors d that satisfy < d < p, so clearly no divisor
CHAMP presentation #1 September 9, 1pm. Ms. Leigh
CHAMP presentation #1 September 9, 1pm Ms. Leigh dog@uh.edu 1 Why do we have all these numbers anyhow? Well the short answer is to solve problems. The longer answer is because mathematicians find them
Discrete Structures Lecture Primes and Greatest Common Divisor
DEFINITION 1 EXAMPLE 1.1 EXAMPLE 1.2 An integer p greater than 1 is called prime if the only positive factors of p are 1 and p. A positive integer that is greater than 1 and is not prime is called composite.
7. Prime Numbers Part VI of PJE
7. Prime Numbers Part VI of PJE 7.1 Definition (p.277) A positive integer n is prime when n > 1 and the only divisors are ±1 and +n. That is D (n) = { n 1 1 n}. Otherwise n > 1 is said to be composite.
Before we talk about prime numbers, we will spend some time with divisibility because there is
Math 1 5.2 Prime Numbers Before we talk about prime numbers, we will spend some time with divisibility. Definition: For whole numbers A and D, with D 0, if there is a whole number Q such that A = D Q,
Primes. Rational, Gaussian, Industrial Strength, etc. Robert Campbell 11/29/2010 1
Primes Rational, Gaussian, Industrial Strength, etc Robert Campbell 11/29/2010 1 Primes and Theory Number Theory to Abstract Algebra History Euclid to Wiles Computation pencil to supercomputer Practical
Intermediate Math Circles March 6, 2013 Number Theory I
What is Number Theory? Intermediate Math Circles March 6, 01 Number Theory I A branch of mathematics where mathematicians examine and study patterns found within the natural number set (positive integers).
Prime Numbers. Prime Numbers. Ramesh Sreekantan ISI, Bangalore. December 1, 2014
Prime Numbers Prime Numbers Ramesh Sreekantan ISI, Bangalore December 1, 2014 Introduction - Prime Numbers Prime numbers are the numbers 2,3,5,7... These are natural numbers which cannot be divided by
An integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p.
Chapter 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p. If n > 1
PRIMALITY TEST FOR FERMAT NUMBERS USING QUARTIC RECURRENCE EQUATION. Predrag Terzic Podgorica, Montenegro
PRIMALITY TEST FOR FERMAT NUMBERS USING QUARTIC RECURRENCE EQUATION Predrag Terzic Podgorica, Montenegro pedja.terzic@hotmail.com Abstract. We present deterministic primality test for Fermat numbers, F
. As the binomial coefficients are integers we have that. 2 n(n 1).
Math 580 Homework. 1. Divisibility. Definition 1. Let a, b be integers with a 0. Then b divides b iff there is an integer k such that b = ka. In the case we write a b. In this case we also say a is a factor
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. (a) Circle the prime
Math 412: Number Theory Lecture 3: Prime Decomposition of
Math 412: Number Theory Lecture 3: Prime Decomposition of Integers Gexin Yu gyu@wm.edu College of William and Mary Prime numbers Definition: a (positive) integer p is prime if p has no divisor other than
CHAPTER 6. Prime Numbers. Definition and Fundamental Results
CHAPTER 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results 6.1. Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and the only positive divisors of p are 1 and p. If n
Math 5330 Spring Notes Prime Numbers
Math 5330 Sring 208 Notes Prime Numbers The study of rime numbers is as old as mathematics itself. This set of notes has a bunch of facts about rimes, or related to rimes. Much of this stuff is old dating
THE FORMATION OF PRIME NUMBERS THE SOLUTION FOR GOLDBACH S CONJECTURES
Research article THE FORMATION OF PRIME NUMBERS AND THE SOLUTION FOR GOLDBACH S CONJECTURES JOSÉ WILLIAM PORRAS-FERREIRA CARLOS ALBERTO ANDRADE Escuela Naval de Cadetes Almirante Padilla, Isla Manzanillo,
Definition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively
6 Prime Numbers Part VI of PJE 6.1 Fundamental Results Definition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively D (p) = { p 1 1 p}. Otherwise
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number
Math 319 Problem Set #2 Solution 14 February 2002
Math 39 Problem Set # Solution 4 February 00. (.3, problem 8) Let n be a positive integer, and let r be the integer obtained by removing the last digit from n and then subtracting two times the digit ust
Disproof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Disproof Fall / 16
Disproof MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Disproof Fall 2014 1 / 16 Outline 1 s 2 Disproving Universal Statements: Counterexamples 3 Disproving Existence
Junior Villafana. Math 301. Dr. Meredith. Odd Perfect Numbers
Junior Villafana Math 301 Dr. Meredith Odd Perfect Numbers Arguably the oldest unsolved problem in mathematics is still giving mathematicians a headache, even with the aid of technology; the search for
Primality testing: then and now
Primality testing: then and now Mathematics Department Colloquium Boise State University, February 20, 2019 Carl Pomerance Dartmouth College (emeritus) University of Georgia (emeritus) In 1801, Carl Friedrich
The first dynamical system
The first dynamical system Carl Pomerance, Dartmouth College Charles University November 8, 2016 As we all know, functions in mathematics are ubiquitous and indispensable. But what was the very first function
MATH 25 CLASS 8 NOTES, OCT
MATH 25 CLASS 8 NOTES, OCT 7 20 Contents. Prime number races 2. Special kinds of prime numbers: Fermat and Mersenne numbers 2 3. Fermat numbers 3. Prime number races We proved that there were infinitely
Introduction Integers. Discrete Mathematics Andrei Bulatov
Introduction Integers Discrete Mathematics Andrei Bulatov Discrete Mathematics - Integers 9- Integers God made the integers; all else is the work of man Leopold Kroenecker Discrete Mathematics - Integers
Lucas Lehmer primality test - Wikipedia, the free encyclopedia
Lucas Lehmer primality test From Wikipedia, the free encyclopedia In mathematics, the Lucas Lehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Edouard Lucas
Elementary Number Theory
Elementary Number Theory A revision by Jim Hefferon, St Michael s College, 2003-Dec of notes by W. Edwin Clark, University of South Florida, 2002-Dec L A TEX source compiled on January 5, 2004 by Jim Hefferon,
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Spring 2006
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 1 / 1 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 2.4 2.6 of Rosen Introduction I When talking
Algorithmic number theory. Questions/Complaints About Homework? The division algorithm. Division
Questions/Complaints About Homework? Here s the procedure for homework questions/complaints: 1. Read the solutions first. 2. Talk to the person who graded it (check initials) 3. If (1) and (2) don t work,
On the Cardinality of Mersenne Primes
On the Cardinality of Mersenne Primes Garimella Rama Murthy, Associate Professor, International Institute of Information Technology (IIIT), Gachibowli, Hyderabad-32,AP,INDIA ABSTRACT In this research paper,
4.3 - Primes and Greatest Common Divisors
4.3 - Primes and Greatest Common Divisors Introduction We focus on properties of integers and prime factors Primes Definition 1 An integer p greater than 1 is called prime if the only positive factors
Math 3336 Section 4.3 Primes and Greatest Common Divisors. Prime Numbers and their Properties Conjectures and Open Problems About Primes
Math 3336 Section 4.3 Primes and Greatest Common Divisors Prime Numbers and their Properties Conjectures and Open Problems About Primes Greatest Common Divisors and Least Common Multiples The Euclidian
Primality Testing- Is Randomization worth Practicing?
Primality Testing- Is Randomization worth Practicing? Shubham Sahai Srivastava Indian Institute of Technology, Kanpur ssahai@cse.iitk.ac.in April 5, 2014 Shubham Sahai Srivastava (IITK) Primality Test
Primality testing: then and now
Seventy-five years of Mathematics of Computation ICERM, November 1 3, 2018 Primality testing: then and now Carl Pomerance Dartmouth College, Emeritus University of Georgia, Emeritus In 1801, Carl Friedrich
5.1. EXAMPLE 1 Decide whether the first number listed is divisible by the second. (a) 45; 9
5.1 Prime and Composite Numbers In earlier chapters we discussed the set of natural numbers, also called the counting numbers or the positive integers: 1, 2, 3,. Number theory deals with the study of the
Divisibility. 1.1 Foundations
1 Divisibility 1.1 Foundations The set 1, 2, 3,...of all natural numbers will be denoted by N. There is no need to enter here into philosophical questions concerning the existence of N. It will suffice
With Question/Answer Animations. Chapter 4
With Question/Answer Animations Chapter 4 Chapter Motivation Number theory is the part of mathematics devoted to the study of the integers and their properties. Key ideas in number theory include divisibility
Elementary Number Theory
Elementary Number Theory 21.8.2013 Overview The course discusses properties of numbers, the most basic mathematical objects. We are going to follow the book: David Burton: Elementary Number Theory What
Lecture 7. January 15, Since this is an Effective Number Theory school, we should list some effective results. x < π(x) holds for all x 67.
Lecture 7 January 5, 208 Facts about primes Since this is an Effective Number Theory school, we should list some effective results. Proposition. (i) The inequality < π() holds for all 67. log 0.5 (ii)
5.2. Perfect Numbers Divisors of a natural number were covered in Section 5.1.
5.2 Smith Numbers The mathematician Albert Wilansky, when phoning his brother-in-law, Mr. Smith, noticed an interesting property concerning Smith s phone number (493-7775). The number 4,937,775 is composite,
CSE 20 DISCRETE MATH. Winter
CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Today's learning goals Define and use the congruence modulo m equivalence relation Perform computations using modular arithmetic
Contest Number Theory
Contest Number Theory Andre Kessler December 7, 2008 Introduction Number theory is one of the core subject areas of mathematics. It can be somewhat loosely defined as the study of the integers. Unfortunately,
Some multiplicative arithmetical functions
Some multiplicative arithmetical functions an invitation to number theory K. N. Raghavan http://www.imsc.res.in/ knr/ IMSc, Chennai August 203 Standard form of prime factorization of a number; GCD and
Number Theory. Jason Filippou UMCP. ason Filippou UMCP)Number Theory History & Definitions / 1
Number Theory Jason Filippou CMSC250 @ UMCP 06-08-2016 ason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 1 / 1 Outline ason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions
Heuristics for Prime Statistics Brown Univ. Feb. 11, K. Conrad, UConn
Heuristics for Prime Statistics Brown Univ. Feb., 2006 K. Conrad, UConn Two quotes about prime numbers Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers,
An Algorithm for Prime Factorization
An Algorithm for Prime Factorization Fact: If a is the smallest number > 1 that divides n, then a is prime. Proof: By contradiction. (Left to the reader.) A multiset is like a set, except repetitions are
A proof of strong Goldbach conjecture and twin prime conjecture
A proof of strong Goldbach conjecture and twin prime conjecture Pingyuan Zhou E-mail:zhoupingyuan49@hotmail.com Abstract In this paper we give a proof of the strong Goldbach conjecture by studying limit
Solutions of the Diophantine Equation 2 x + p y = z 2 When p is Prime
Annals of Pure and Applied Mathematics Vol. 16, No. 2, 2018, 471-477 ISSN: 2279-087X (P), 2279-0888(online) Published on 29 March 2018 www.researchmathsci.org DOI: http://dx.doi.org/10.22457/apam.v16n2a25
SQUARE PATTERNS AND INFINITUDE OF PRIMES
SQUARE PATTERNS AND INFINITUDE OF PRIMES KEITH CONRAD 1. Introduction Numerical data suggest the following patterns for prime numbers p: 1 mod p p = 2 or p 1 mod 4, 2 mod p p = 2 or p 1, 7 mod 8, 2 mod
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
Fermat's Little Theorem: Proofs That Fermat Might Have Used Author(s): Bob Burn Source: The Mathematical Gazette, Vol. 86, No. 507 (Nov., 2002), pp. 415-422 Published by: The Mathematical Association Stable
What do we actually know about prime numbers distribution?
1 On the number of composite numbers less than a given value. Lemmas, continued. Paper III: What do we know about prime numbers distribution? Paper II presented 3 of 7 lemmas that confirm the conjecture
Appendix: a brief history of numbers
Appendix: a brief history of numbers God created the natural numbers. Everything else is the work of man. Leopold Kronecker (1823 1891) Fundamentals of Computing 2017 18 (2, appendix) http://www.dcs.bbk.ac.uk/~michael/foc/foc.html
Sifting the Primes. Gihan Marasingha University of Oxford
Sifting the Primes Gihan Marasingha University of Oxford 18 March 2005 Irreducible forms: q 1 (x, y) := a 1 x 2 + 2b 1 xy + c 1 y 2, q 2 (x, y) := a 2 x 2 + 2b 2 xy + c 2 y 2, a i, b i, c i Z. Variety
Introduction to Number Theory
INTRODUCTION Definition: Natural Numbers, Integers Natural numbers: N={0,1,, }. Integers: Z={0,±1,±, }. Definition: Divisor If a Z can be writeen as a=bc where b, c Z, then we say a is divisible by b or,
Sundaram's Sieve. by Julian Havil. Sundaram's Sieve
1997 2009, Millennium Mathematics Project, University of Cambridge. Permission is granted to print and copy this page on paper for non commercial use. For other uses, including electronic redistribution,
Introduction to Sets and Logic (MATH 1190)
Introduction to Sets and Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University Nov 13, 2014 Quiz announcement The second quiz will be held on Thursday,
10 Problem 1. The following assertions may be true or false, depending on the choice of the integers a, b 0. a "
Math 4161 Dr. Franz Rothe December 9, 2013 13FALL\4161_fall13f.tex Name: Use the back pages for extra space Final 70 70 Problem 1. The following assertions may be true or false, depending on the choice
On the Prime Divisors of Odd Perfect Numbers
On the Prime Divisors of Odd Perfect Numbers Justin Sweeney Department of Mathematics Trinity College Hartford, CT justin.sweeney@trincoll.edu April 27, 2009 1 Contents 1 History of Perfect Numbers 5 2
Corollary 4.2 (Pepin s Test, 1877). Let F k = 2 2k + 1, the kth Fermat number, where k 1. Then F k is prime iff 3 F k 1
4. Primality testing 4.1. Introduction. Factorisation is concerned with the problem of developing efficient algorithms to express a given positive integer n > 1 as a product of powers of distinct primes.
Integer and Polynomial Arithmetic First Steps towards Abstract Algebra. Arjeh M. Cohen Hans Cuypers Hans Sterk
Integer and Polynomial Arithmetic First Steps towards Abstract Algebra Arjeh M. Cohen Hans Cuypers Hans Sterk October 8, 2007 2 Contents 1 Integer arithmetic 7 1.1 Divisors and multiples......................
Wednesday, February 21. Today we will begin Course Notes Chapter 5 (Number Theory).
Wednesday, February 21 Today we will begin Course Notes Chapter 5 (Number Theory). 1 Return to Chapter 5 In discussing Methods of Proof (Chapter 3, Section 2) we introduced the divisibility relation from
Master Map Algebra (Master) Content Skills Assessment Instructional Strategies Notes. A. MM Quiz 1. A. MM Quiz 2 A. MM Test
Blair High School Teacher: Brent Petersen Master Map Algebra (Master) August 2010 Connections to Algebra Learn how to write and -Variables in Algebra -Exponents and Powers -Order of Operations -Equations
8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
Cryptography. Number Theory with AN INTRODUCTION TO. James S. Kraft. Lawrence C. Washington. CRC Press
AN INTRODUCTION TO Number Theory with Cryptography James S Kraft Gilman School Baltimore, Maryland, USA Lawrence C Washington University of Maryland College Park, Maryland, USA CRC Press Taylor & Francis
SIX PROOFS OF THE INFINITUDE OF PRIMES
SIX PROOFS OF THE INFINITUDE OF PRIMES ALDEN MATHIEU 1. Introduction The question of how many primes exist dates back to at least ancient Greece, when Euclid proved the infinitude of primes (circa 300
Students will be able to simplify numerical expressions and evaluate algebraic expressions. (M)
Morgan County School District Re-3 August What is algebra? This chapter develops some of the basic symbolism and terminology that students may have seen before but still need to master. The concepts of
On the Composite Terms in Sequence Generated from Mersenne-type Recurrence Relations
On the Composite Terms in Sequence Generated from Mersenne-type Recurrence Relations Pingyuan Zhou E-mail:zhoupingyuan49@hotmail.com Abstract We conjecture that there is at least one composite term in
ARITHMETIC OF POSITIVE INTEGERS HAVING PRIME SUMS OF COMPLEMENTARY DIVISORS
Math. J. Okayama Univ. 60 (2018), 155 164 ARITHMETIC OF POSITIVE INTEGERS HAVING PRIME SUMS OF COMPLEMENTARY DIVISORS Kenichi Shimizu Abstract. We study a class of integers called SP numbers (Sum Prime
Adam Blank Spring 2017 CSE 311. Foundations of Computing I. * All slides are a combined effort between previous instructors of the course
Adam Blank Spring 2017 CSE 311 Foundations of Computing I * All slides are a combined effort between previous instructors of the course HW 3 De-Brief HW 3 De-Brief PROOFS! HW 3 De-Brief Proofs This is
August 2018 ALGEBRA 1
August 0 ALGEBRA 3 0 3 Access to Algebra course :00 Algebra Orientation Course Introduction and Reading Checkpoint 0.0 Expressions.03 Variables.0 3.0 Translate Words into Variable Expressions DAY.0 Translate
Big doings with small g a p s
Big doings with small g a p s Paul Pollack University of Georgia AAAS Annual Meeting February 16, 2015 1 of 26 SMALL GAPS: A SHORT SURVEY 2 of 26 300 BCE Let p n denote the nth prime number, so p 1 = 2,
The ranges of some familiar arithmetic functions
The ranges of some familiar arithmetic functions Max-Planck-Institut für Mathematik 2 November, 2016 Carl Pomerance, Dartmouth College Let us introduce our cast of characters: ϕ, λ, σ, s Euler s function:
Distribution of Prime Numbers Prime Constellations Diophantine Approximation. Prime Numbers. How Far Apart Are They? Stijn S.C. Hanson.
Distribution of How Far Apart Are They? June 13, 2014 Distribution of 1 Distribution of Behaviour of π(x) Behaviour of π(x; a, q) 2 Distance Between Neighbouring Primes Beyond Bounded Gaps 3 Classical
ENTRY NUMBER THEORY. [ENTRY NUMBER THEORY] Authors: Oliver Knill: 2003 Literature: Hua, introduction to number theory.
![ENTRY NUMBER THEORY. [ENTRY NUMBER THEORY] Authors: Oliver Knill: 2003 Literature: Hua, introduction to number theory.](/thumbs/74/70686567.jpg "ENTRY NUMBER THEORY. [ENTRY NUMBER THEORY] Authors: Oliver Knill: 2003 Literature: Hua, introduction to number theory.")
ENTRY NUMBER THEORY [ENTRY NUMBER THEORY] Authors: Oliver Knill: 2003 Literature: Hua, introduction to number theory ABC conjecture [ABC conjecture] If a,b,c are positive integers, let N(a,b,c) be the
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
On Runs of Residues Author(s): D. H. Lehmer and Emma Lehmer Source: Proceedings of the American Mathematical Society, Vol. 13, No. 1 (Feb., 1962), pp. 102-106 Published by: American Mathematical Society
History of Mathematics
History of Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Summer 2011 6A: Fermat: Beginning of modern number theory (2) Fermat s little theorem Given a prime p, and any geometric
1. multiplication is commutative and associative;
Chapter 4 The Arithmetic of Z In this chapter, we start by introducing the concept of congruences; these are used in our proof (going back to Gauss 1 ) that every integer has a unique prime factorization.
The Riddle of Primes
A talk given at Dalian Univ. of Technology (Nov. 16, 2012) and Nankai University (Dec. 1, 2012) The Riddle of Primes Zhi-Wei Sun Nanjing University Nanjing 210093, P. R. China zwsun@nju.edu.cn http://math.nju.edu.cn/
p = This is small enough that its primality is easily verified by trial division. A candidate prime above 1000 p of the form p U + 1 is
LARGE PRIME NUMBERS 1. Fermat Pseudoprimes Fermat s Little Theorem states that for any positive integer n, if n is prime then b n % n = b for b = 1,..., n 1. In the other direction, all we can say is that
(Primes and) Squares modulo p
(Primes and) Squares modulo p Paul Pollack MAA Invited Paper Session on Accessible Problems in Modern Number Theory January 13, 2018 1 of 15 Question Consider the infinite arithmetic progression Does it
Odd Primes. Theory W: Structure and organization of odd primes (8th problem of Hilbert) mathscience.tsoft .com
T-Soft@mail mathscience.tsoftemail.com Odd Primes Theory W: Structure and organization of odd primes (8th problem of Hilbert) WOLF François and WOLF Marc PhDs in materials science Ceramists engineers IT
Verbatim copying and redistribution of this document. are permitted in any medium provided this notice and. the copyright notice are preserved.
New Results on Primes from an Old Proof of Euler s by Charles W. Neville CWN Research 55 Maplewood Ave. West Hartford, CT 06119, U.S.A. cwneville@cwnresearch.com September 25, 2002 Revision 1, April, 2003
The Riemann Hypothesis
The Riemann Hypothesis Matilde N. Laĺın GAME Seminar, Special Series, History of Mathematics University of Alberta mlalin@math.ualberta.ca http://www.math.ualberta.ca/~mlalin March 5, 2008 Matilde N. Laĺın
PRIME NUMBERS YANKI LEKILI
PRIME NUMBERS YANKI LEKILI We denote by N the set of natural numbers: 1,2,..., These are constructed using Peano axioms. We will not get into the philosophical questions related to this and simply assume
Goldbach Conjecture: An invitation to Number Theory by R. Balasubramanian Institute of Mathematical Sciences, Chennai
Goldbach Conjecture: An invitation to Number Theory by R. Balasubramanian Institute of Mathematical Sciences, Chennai balu@imsc.res.in R. Balasubramanian (IMSc.) Goldbach Conjecture 1 / 22 Goldbach Conjecture:
[Part 2] Asymmetric-Key Encipherment. Chapter 9. Mathematics of Cryptography. Objectives. Contents. Objectives
![[Part 2] Asymmetric-Key Encipherment. Chapter 9. Mathematics of Cryptography. Objectives. Contents. Objectives](/thumbs/75/72408741.jpg "[Part 2] Asymmetric-Key Encipherment. Chapter 9. Mathematics of Cryptography. Objectives. Contents. Objectives")
[Part 2] Asymmetric-Key Encipherment Mathematics of Cryptography Forouzan, B.A. Cryptography and Network Security (International Edition). United States: McGraw Hill, 2008. Objectives To introduce prime
Prime Number Theory and the Riemann Zeta-Function
5262589 - Recent Perspectives in Random Matrix Theory and Number Theory Prime Number Theory and the Riemann Zeta-Function D.R. Heath-Brown Primes An integer p N is said to be prime if p and there is no
Fermat s Little Theorem. Fermat s little theorem is a statement about primes that nearly characterizes them.
Fermat s Little Theorem Fermat s little theorem is a statement about primes that nearly characterizes them. Theorem: Let p be prime and a be an integer that is not a multiple of p. Then a p 1 1 (mod p).
Polynomial Formula at the origin of the Prime Numbers Generator
Polynomial Formula at the origin of the Prime Numbers Generator WOLF François; WOLF Marc F r a n c e 2 3 a v e n u e o f W a i l l y 7 8 2 9 0 C r o i s s y 06-89- 83-11- 11 f r a n c o i s. w o l f @
Equivalence of Pepin s and the Lucas-Lehmer Tests
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol., No. 3, 009, (35-360) ISSN 1307-5543 www.ejpam.com Equivalence of Pepin s and the Lucas-Lehmer Tests John H. Jaroma Department of Mathematics & Physics,
Proofs of the infinitude of primes
Proofs of the infinitude of primes Tomohiro Yamada Abstract In this document, I would like to give several proofs that there exist infinitely many primes. 0 Introduction It is well known that the number
Elementary Proof That There are Infinitely Many Primes p such that p 1 is a Perfect Square (Landau's Fourth Problem)
Elementary Proof That There are Infinitely Many Primes such that 1 is a Perfect Square (Landau's Fourth Problem) Stehen Marshall 7 March 2017 Abstract This aer resents a comlete and exhaustive roof of
Basic properties of real numbers. Solving equations and inequalities. Homework. Solve and write linear equations.
August Equations and inequalities S. 1.1a,1.2a,1.3a, 2.1a, 2.3 a-c, 6.2a. Simplifying expressions. Algebra II Honors Textbook Glencoe McGraw Hill Algebra II and supplements McDougal Littell Houghton Miffin
Mathematics: Pre-Algebra Eight
Mathematics: Pre-Algebra Eight Students in 8 th Grade Pre-Algebra will study expressions and equations using one or two unknown values. Students will be introduced to the concept of a mathematical function
Primality testing: variations on a theme of Lucas. Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA
Primality testing: variations on a theme of Lucas Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA In 1801, Carl Friedrich Gauss wrote: The problem of distinguishing prime numbers from composite