Repeated Values of Euler s Function. a talk by Paul Kinlaw on joint work with Jonathan Bayless
|
|
- Angelica Gallagher
- 5 years ago
- Views:
Transcription
1 Repeated Values of Euler s Function a talk by Paul Kinlaw on joint work with Jonathan Bayless Oct 4 205
2 Problem of Erdős: Consider solutions of the equations ϕ(n) = ϕ(n + k), σ(n) = σ(n + k) for fixed k N, particularly k =. Currently Unsolved: Are there are infinitely many solutions? Schinzel and Sierpiński (958): The sequence ϕ(n+)/ϕ(n) is dense in (0, ). (Similar result for σ). Erdős-Mirsky Conjecture (Proved by Heath- Brown, 984): There infinitely many n for which d(n + ) = d(n). The same has been proved for ω and Ω.
3 Notation: Let A = {n N : ϕ(n) = ϕ(n + )} and A(x) = #{n x : n A}. Similarly, let B = {n N : σ(n) = σ(n + )} and let B(x) = #{n x : n B}. Theorem (Erdős, 936:) The asymptotic density of n with ϕ(n) < ϕ(n + ) is /2, and the same for ϕ(n) > ϕ(n + ). Thus A has asymptotic density 0. The same holds for B. Theorem (Erdős, Pomerance and Sárközy, 987): A(x) < x/ exp( 3 log x) for sufficiently large x, and the same for B. It follows that n A n <, n B n < They conjecture that A(x) > x ɛ for every ɛ > 0 and x x 0 (ɛ), and similarly for B. 2
4 Theorem (Bayless, K): We have Also, < n A < n B n < n < We have recently improved the upper bounds to 9553 for A and for B. Def: A multiperfect number is a number n such that n σ(n). Let M denote the set of multiperfect numbers and M(x) its counting function. It is conjectured that M is infinite. Theorem (Erdős, 956): M(x) x 0.75+ɛ for all ɛ > 0 and x x 0 (ɛ). Theorem (Bayless, K): n M n =
5 Proposition (Bayless, K): There are infinitely many n such that ϕ(n) = ϕ(n + k) for some k < n The same holds for σ. Proof: Baker and Harman proved that ϕ (m) > m infinitely often. It is also known if ϕ(n) = m then n < 3m log log m < m +ɛ for all ɛ > 0 and sufficiently large m. Thus ϕ (m) [, m +ɛ ] has size greater than m The result follows by the Pigeon Hole principle. 4
6 Proposition (Bayless, K): If ϕ(n) = ϕ(n + ) then ϕ(n) n < 0.5 with precisely six exceptions: n =, 3, 5, 255, 65535,
7 Proposition (Bayless, K): If ϕ(n) = ϕ(n + ) then ϕ(n)/n < 0.5 with precisely six exceptions, when n is the square-free product of the first k Fermat primes, k = 0,..., 5. Sketch of Proof: We have ϕ(n) n = ϕ(n + ) n + ( + n Thus by the product formula for ϕ, p n ( p ) = ( + ) n q n+ ( ). q Consider three cases, n is even, n is odd but n+ is not a power of two, or n+ is a power of two. (The third case yields the 6 exceptions.) ). 6
8 Proposition (Bayless, K): If σ(n) = σ(n + ) then σ(n)/n >.5. Sketch of Proof: Use the estimate p n ( + p ) σ(n) n < p n ( + p valid for n >. Consider three cases: n is odd, n is even but not a power of 2, or n is a power of two (the third case cannot occur). ) 7
9 Explicit Bounds on the sum of reciprocals of n such that ϕ(n) = ϕ(n + ). Small Range: n 0 2. An exhaustive list of all 5236 solutions in this range was computed by Noe and McCranie (OEIS). We compute to obtain n A n 0 2 n =
10 Middle Range: 0 2 < n e 06. We need a very large cutoff to get a good upper bound on the large range. We feared the worst: n < <n e 06 However Carl Pomerance suggested to us that solutions n (mod 3) appear to be rare. Observation: All solutions < n 0 2 are congruent to 2 or 3 (mod 6), with relative frequencies very close to 0.5. We cannot find any exceptions n > 0 2. This led us to the following result. Lemma (Bayless, K): Suppose n > and ϕ(n) = ϕ(n + ). (.) If n 5 (mod 6) then ω(n) 33. (2.) If n 0 (mod 6) then ω(n + ) 33. (3.) If n (mod 6) and gcd(n, 35) = then ω(n) 27. (4.) If n 4 (mod 6) and gcd(n +, 35) = then ω(n + ) 27. 9
11 Explicit Hardy-Ramanujan Inequality Let π k (x) = #{n x : ω(n) = k}. π k (x) x(log log x)k (k )! log x. The same asymptotic formula also applies to: τ k (x) = #{n x : Ω(n) = k}, π sf k (x) = #{n x : ω(n) = Ω(n) = k}, υ k (x) = #{n x : Ω(n) k}. These are variants of Landau s 900 theorem, proved for τ k (x) by induction on k. 0
12 Hardy-Ramanujan Inequality: C, C 2 > 0. π k (x) C x(log log x + C 2 ) k. (k )! log x Theorem (Bayless, Klyve) For x 0 2, π sf k.0924x(log log x )k (x). (k )! log x Theorem (Bayless, K) For x 0 2, π k (x).0989x(log log x +.74)k. (k )! log x This bounds the contribution to the middle range from 8 residue classes mod 20.
13 The Large Range: We follow the proof of Erdős, Pomerance and Sárközy, making all implied constants explicit as we proceed. Five classes of numbers are dealt with separately, and then the count of remaining numbers is bounded. We can then apply partial summation to bound the contribution to the reciprocal sum. For instance, their proof uses P (n) L 2, where ( ) L(x) = exp 8 (log x)2/3 log log x and addresses the count of L 2 -smooth numbers n (i.e. P (n) L 2 ) up to x separately. 2
14 De Bruijn s Ψ function: Ψ(x, y) = #{n x : P (n) y}. Theorem (de Bruijn): There exists c > 0 such that Ψ(x, y) x exp( cu log u) where u = log x/ log y. Lemma (Bayless, K): For x 0 2 we have Ψ(x, y).033x exp( 0.7u log u). Conclusion: Our best bound on the reciprocal sum comes from this improvement (addition of the log u term above), and by adjusting the cutoffs for two different large ranges, as well as other coefficients. 3
15 Some open problems. Improve the bounds on these reciprocal sums. Do there exist solutions n > of the equation ϕ(n) = ϕ(n + ) that are not 2 or 3 mod 6? 4
16 Here are the five assumptions made in the proof of Pomerance, Erdős and Sárközy. Counts of integers violating the assumptions are bounded separately. Here and L(x) = exp l(x) = exp( 3 log x) ( ) 8 (log x)2/3 log log x.. P (n) L 2 and P (n + ) L If k a divides n or n + where a 2, then k a l min{n/p (n), (n + )/P (n + )} L. 4. min{p (ϕ(m)), P (ϕ(m ))} l 4, for m = n/p (n) and m = (n + )/P (n + ). 5. P (n) > P (n + ). 5
Dense product-free sets of integers
Dense product-free sets of integers Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA Joint Mathematics Meetings Boston, January 6, 2012 Based on joint work with P. Kurlberg, J. C. Lagarias,
More informationAmicable numbers. CRM Workshop on New approaches in probabilistic and multiplicative number theory December 8 12, 2014
Amicable numbers CRM Workshop on New approaches in probabilistic and multiplicative number theory December 8 12, 2014 Carl Pomerance, Dartmouth College (U. Georgia, emeritus) Recall that s(n) = σ(n) n,
More informationFlat primes and thin primes
Flat primes and thin primes Kevin A. Broughan and Zhou Qizhi University of Waikato, Hamilton, New Zealand Version: 0th October 2008 E-mail: kab@waikato.ac.nz, qz49@waikato.ac.nz Flat primes and thin primes
More informationEuler s ϕ function. Carl Pomerance Dartmouth College
Euler s ϕ function Carl Pomerance Dartmouth College Euler s ϕ function: ϕ(n) is the number of integers m [1, n] with m coprime to n. Or, it is the order of the unit group of the ring Z/nZ. Euler: If a
More informationSums and products. Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA. Dartmouth Mathematics Society May 16, 2012
Sums and products Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA Dartmouth Mathematics Society May 16, 2012 Based on joint work with P. Kurlberg, J. C. Lagarias, & A. Schinzel Let s begin
More informationCounting subgroups of the multiplicative group
Counting subgroups of the multiplicative group Lee Troupe joint w/ Greg Martin University of British Columbia West Coast Number Theory 2017 Question from I. Shparlinski to G. Martin, circa 2009: How many
More informationStatistics in elementary number theory
Statistics in elementary number theory Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA CRM Workshop on Statistics and Number Theory 15 19 September, 2014 In this talk we will consider 4 statistical
More informationHOW OFTEN IS EULER S TOTIENT A PERFECT POWER? 1. Introduction
HOW OFTEN IS EULER S TOTIENT A PERFECT POWER? PAUL POLLACK Abstract. Fix an integer k 2. We investigate the number of n x for which ϕn) is a perfect kth power. If we assume plausible conjectures on the
More informationA 1935 Erdős paper on prime numbers and Euler s function
A 1935 Erdős paper on prime numbers and Euler s function Carl Pomerance, Dartmouth College with Florian Luca, UNAM, Morelia 1 2 3 4 Hardy & Ramanujan, 1917: The normal number of prime divisors of n is
More informationThe 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:
More informationTwo problems in combinatorial number theory
Two problems in combinatorial number theory Carl Pomerance, Dartmouth College Debrecen, Hungary October, 2010 Our story begins with: Abram S. Besicovitch 1 Besicovitch showed in 1934 that there are primitive
More informationSums and products. Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA
Sums and products Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA International Number Theory Conference in Memory of Alf van der Poorten, AM 12 16 March, 2012 CARMA, the University of Newcastle
More informationThe ranges of some familiar arithmetic functions
The ranges of some familiar arithmetic functions Carl Pomerance Dartmouth College, emeritus University of Georgia, emeritus based on joint work with K. Ford, F. Luca, and P. Pollack and T. Freiburg, N.
More informationf(n) = f(p e 1 1 )...f(p e k k ).
3. Arithmetic functions 3.. Arithmetic functions. These are functions f : N N or Z or maybe C, usually having some arithmetic significance. An important subclass of such functions are the multiplicative
More informationTHE RECIPROCAL SUM OF THE AMICABLE NUMBERS
THE RECIPROCAL SUM OF THE AMICABLE NUMBERS HANH MY NGUYEN AND CARL POMERANCE ABSTRACT. In this paper, we improve on several earlier attempts to show that the reciprocal sum of the amicable numbers is small,
More informationThe Chinese Remainder Theorem
Chapter 5 The Chinese Remainder Theorem 5.1 Coprime moduli Theorem 5.1. Suppose m, n N, and gcd(m, n) = 1. Given any remainders r mod m and s mod n we can find N such that N r mod m and N s mod n. Moreover,
More informationThe 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
More informationTHE RECIPROCAL SUM OF THE AMICABLE NUMBERS
THE RECIPROCAL SUM OF THE AMICABLE NUMBERS HANH MY NGUYEN AND CARL POMERANCE ABSTRACT. In this paper, we improve on several earlier attempts to show that the reciprocal sum of the amicable numbers is small,
More informationThe ranges of some familiar functions
The ranges of some familiar functions CRM Workshop on New approaches in probabilistic and multiplicative number theory December 8 12, 2014 Carl Pomerance, Dartmouth College (U. Georgia, emeritus) Let us
More informationOn the Distribution of Carmichael Numbers
On the Distribution of Carmichael Numbers Aran Nayebi arxiv:0906.3533v10 [math.nt] 12 Aug 2009 Abstract {1+o(1)} log log log x 1 Pomerance conjectured that there are x log log x Carmichael numbers up to
More informationDartmouth College. The Reciprocal Sum of the Amicable Numbers
Dartmouth College Hanover, New Hampshire The Reciprocal Sum of the Amicable Numbers A thesis submitted in partial fulfillment of the requirements for the degree Bachelor of Arts in Mathematics by Hanh
More informationSums and products. Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA. Providence College Math/CS Colloquium April 2, 2014
Sums and products Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA Providence College Math/CS Colloquium April 2, 2014 Let s begin with products. Take the N N multiplication table. It has
More informationOn Carmichael numbers in arithmetic progressions
On Carmichael numbers in arithmetic progressions William D. Banks Department of Mathematics University of Missouri Columbia, MO 65211 USA bbanks@math.missouri.edu Carl Pomerance Department of Mathematics
More informationPlaying Ball with the Largest Prime Factor
Playing Ball with the Largest Prime Factor An Introduction to Ruth-Aaron Numbers Madeleine Farris Wellesley College July 30, 2018 The Players The Players Figure: Babe Ruth Home Run Record: 714 The Players
More informationSums of distinct divisors
Oberlin College February 3, 2017 1 / 50 The anatomy of integers 2 / 50 What does it mean to study the anatomy of integers? Some natural problems/goals: Study the prime factors of integers, their size and
More informationON UNTOUCHABLE NUMBERS AND RELATED PROBLEMS
ON UNTOUCHABLE NUMBERS AND RELATED PROBLEMS CARL POMERANCE AND HEE-SUNG YANG Abstract. In 1973, Erdős proved that a positive proportion of numbers are untouchable, that is, not of the form σ(n) n, the
More informationSums and products: What we still don t know about addition and multiplication
Sums and products: What we still don t know about addition and multiplication Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA Based on joint work with R. P. Brent, P. Kurlberg, J. C. Lagarias,
More informationSums of Squares. Bianca Homberg and Minna Liu
Sums of Squares Bianca Homberg and Minna Liu June 24, 2010 Abstract For our exploration topic, we researched the sums of squares. Certain properties of numbers that can be written as the sum of two squares
More informationOn a problem of Nicol and Zhang
Journal of Number Theory 28 2008 044 059 www.elsevier.com/locate/jnt On a problem of Nicol and Zhang Florian Luca a, József Sándor b, a Instituto de Matemáticas, Universidad Nacional Autonoma de México,
More informationProofs 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
More informationCarmichael numbers and the sieve
June 9, 2015 Dedicated to Carl Pomerance in honor of his 70th birthday Carmichael numbers Fermat s little theorem asserts that for any prime n one has a n a (mod n) (a Z) Carmichael numbers Fermat s little
More informationIrreducible radical extensions and Euler-function chains
Irreducible radical extensions and Euler-function chains Florian Luca Carl Pomerance June 14, 2006 For Ron Graham on his 70th birthday Abstract We discuss the smallest algebraic number field which contains
More informationAN UPPER BOUND FOR ODD PERFECT NUMBERS. Pace P. Nielsen Department of Mathematics, University of California, Berkeley, CA 94720, U.S.A.
AN UPPER BOUND FOR ODD PERFECT NUMBERS Pace P. Nielsen Department of Mathematics, University of California, Berkeley, CA 94720, U.S.A. pace@math.berkeley.edu Received:1/31/03, Revised: 9/29/03, Accepted:
More informationOn the parity of k-th powers modulo p
On the parity of k-th powers modulo p Jennifer Paulhus Kansas State University paulhus@math.ksu.edu www.math.ksu.edu/ paulhus This is joint work with Todd Cochrane and Chris Pinner of Kansas State University
More informationE-SYMMETRIC NUMBERS (PUBLISHED: COLLOQ. MATH., 103(2005), NO. 1, )
E-SYMMETRIC UMBERS PUBLISHED: COLLOQ. MATH., 032005), O., 7 25.) GAG YU Abstract A positive integer n is called E-symmetric if there exists a positive integer m such that m n = φm), φn)). n is called E-asymmetric
More informationCarmichael numbers with a totient of the form a 2 + nb 2
Carmichael numbers with a totient of the form a 2 + nb 2 William D. Banks Department of Mathematics University of Missouri Columbia, MO 65211 USA bankswd@missouri.edu Abstract Let ϕ be the Euler function.
More informationINTEGERS DIVISIBLE BY THE SUM OF THEIR PRIME FACTORS
INTEGERS DIVISIBLE BY THE SUM OF THEIR PRIME FACTORS JEAN-MARIE DE KONINCK and FLORIAN LUCA Abstract. For each integer n 2, let β(n) be the sum of the distinct prime divisors of n and let B(x) stand for
More informationGod 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
More informationThe first function. Carl Pomerance, Dartmouth College (U. Georgia, emeritus)
The first function Carl Pomerance, Dartmouth College (U. Georgia, emeritus) Connections in Discrete Mathematics A celebration of the work of Ron Graham Simon Fraser University, June 15 19, 2015 On a papyrus
More informationOn pseudosquares and pseudopowers
On pseudosquares and pseudopowers Carl Pomerance Department of Mathematics Dartmouth College Hanover, NH 03755-3551, USA carl.pomerance@dartmouth.edu Igor E. Shparlinski Department of Computing Macquarie
More informationON CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS
J. Aust. Math. Soc. 88 (2010), 313 321 doi:10.1017/s1446788710000169 ON CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS WILLIAM D. BANKS and CARL POMERANCE (Received 4 September 2009; accepted 4 January
More informationOn the number of co-prime-free sets. Neil J. Calkin and Andrew Granville *
On the number of co-prime-free sets. by Neil J. Calkin and Andrew Granville * Abstract: For a variety of arithmetic properties P such as the one in the title) we investigate the number of subsets of the
More informationLecture 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)
More informationOn pseudosquares and pseudopowers
On pseudosquares and pseudopowers Carl Pomerance Department of Mathematics Dartmouth College Hanover, NH 03755-3551, USA carl.pomerance@dartmouth.edu Igor E. Shparlinski Department of Computing Macquarie
More informationDepartmento de Matemáticas, Universidad de Oviedo, Oviedo, Spain
#A37 INTEGERS 12 (2012) ON K-LEHMER NUMBERS José María Grau Departmento de Matemáticas, Universidad de Oviedo, Oviedo, Spain grau@uniovi.es Antonio M. Oller-Marcén Centro Universitario de la Defensa, Academia
More informationSchool of Mathematics
School of Mathematics Programmes in the School of Mathematics Programmes including Mathematics Final Examination Final Examination 06 22498 MSM3P05 Level H Number Theory 06 16214 MSM4P05 Level M Number
More informationODD REPDIGITS TO SMALL BASES ARE NOT PERFECT
#A34 INTEGERS 1 (01) ODD REPDIGITS TO SMALL BASES ARE NOT PERFECT Kevin A. Broughan Department of Mathematics, University of Waikato, Hamilton 316, New Zealand kab@waikato.ac.nz Qizhi Zhou Department of
More informationarxiv:math/ v3 [math.nt] 23 Apr 2004
Complexity of Inverting the Euler Function arxiv:math/0404116v3 [math.nt] 23 Apr 2004 Scott Contini Department of Computing Macquarie University Sydney, NSW 2109, Australia contini@ics.mq.edu.au Igor E.
More informationExplicit Bounds for the Sum of Reciprocals of Pseudoprimes and Carmichael Numbers
2 3 47 6 23 Journal of Integer Sequences, Vol. 20 207), Article 7.6.4 Exlicit Bounds for the Sum of Recirocals of Pseudorimes and Carmichael Numbers Jonathan Bayless and Paul Kinlaw Husson University College
More informationProposed by Jean-Marie De Koninck, Université Laval, Québec, Canada. (a) Let φ denote the Euler φ function, and let γ(n) = p n
10966. Proposed by Jean-Marie De Koninck, Université aval, Québec, Canada. (a) et φ denote the Euler φ function, and let γ(n) = p n p, with γ(1) = 1. Prove that there are exactly six positive integers
More informationMath 259: Introduction to Analytic Number Theory The Selberg (quadratic) sieve and some applications
Math 259: Introduction to Analytic Number Theory The Selberg (quadratic) sieve and some applications An elementary and indeed naïve approach to the distribution of primes is the following argument: an
More informationSome problems of Erdős on the sum-of-divisors function
Some problems of Erdős on the sum-of-divisors function Paul Pollack Cleveland Area Number Theory Seminar March 11, 2014 1 of 43 Dramatis Personae Let s(n) := d n,d
More informationPALINDROMIC SUMS OF PROPER DIVISORS. Paul Pollack Department of Mathematics, University of Georgia, Athens, GA 30602, USA
PALINDROMIC SUMS OF PROPER DIVISORS Paul Pollack Department of Mathematics, University of Georgia, Athens, GA 30602, USA pollack@math.uga.edu Received:, Revised:, Accepted:, Published: Abstract Fi an integer
More informationSOLUTIONS Math 345 Homework 6 10/11/2017. Exercise 23. (a) Solve the following congruences: (i) x (mod 12) Answer. We have
Exercise 23. (a) Solve the following congruences: (i) x 101 7 (mod 12) Answer. We have φ(12) = #{1, 5, 7, 11}. Since gcd(7, 12) = 1, we must have gcd(x, 12) = 1. So 1 12 x φ(12) = x 4. Therefore 7 12 x
More informationChapter 1. Introduction to prime number theory. 1.1 The Prime Number Theorem
Chapter 1 Introduction to prime number theory 1.1 The Prime Number Theorem In the first part of this course, we focus on the theory of prime numbers. We use the following notation: we write f( g( as if
More informationLARGE PRIME NUMBERS (32, 42; 4) (32, 24; 2) (32, 20; 1) ( 105, 20; 0).
LARGE PRIME NUMBERS 1. Fast Modular Exponentiation Given positive integers a, e, and n, the following algorithm quickly computes the reduced power a e % n. (Here x % n denotes the element of {0,, n 1}
More informationElementary Number Theory Review. Franz Luef
Elementary Number Theory Review Principle of Induction Principle of Induction Suppose we have a sequence of mathematical statements P(1), P(2),... such that (a) P(1) is true. (b) If P(k) is true, then
More informationNotes on Primitive Roots Dan Klain
Notes on Primitive Roots Dan Klain last updated March 22, 2013 Comments and corrections are welcome These supplementary notes summarize the presentation on primitive roots given in class, which differed
More informationNumber Theory Homework.
Number Theory Homewor. 1. The Theorems of Fermat, Euler, and Wilson. 1.1. Fermat s Theorem. The following is a special case of a result we have seen earlier, but as it will come up several times in this
More informationCounting in number theory. Fibonacci integers. Carl Pomerance, Dartmouth College. Rademacher Lecture 3, University of Pennsylvania September, 2010
Counting in number theory Fibonacci integers Carl Pomerance, Dartmouth College Rademacher Lecture 3, University of Pennsylvania September, 2010 Leonardo of Pisa (Fibonacci) (c. 1170 c. 1250) 1 We all know
More informationAnalytic. Number Theory. Exploring the Anatomy of Integers. Jean-Marie. De Koninck. Florian Luca. ffk li? Graduate Studies.
Analytic Number Theory Exploring the Anatomy of Integers Jean-Marie Florian Luca De Koninck Graduate Studies in Mathematics Volume 134 ffk li? American Mathematical Society Providence, Rhode Island Preface
More informationContents. Preface to the First Edition. Preface to the Second Edition. Preface to the Third Edition
Contents Preface to the First Edition Preface to the Second Edition Preface to the Third Edition i iii iv Glossary of Symbols A. Prime Numbers 3 A1. Prime values of quadratic functions. 7 A2. Primes connected
More informationMAS114: Exercises. October 26, 2018
MAS114: Exercises October 26, 2018 Note that the challenge problems are intended to be difficult! Doing any of them is an achievement. Please hand them in on a separate piece of paper if you attempt them.
More informationOn the number of elements with maximal order in the multiplicative group modulo n
ACTA ARITHMETICA LXXXVI.2 998 On the number of elements with maximal order in the multiplicative group modulo n by Shuguang Li Athens, Ga.. Introduction. A primitive root modulo the prime p is any integer
More informationPerfect numbers among polynomial values
Perfect numbers among polynomial values Oleksiy Klurman Université de Montréal June 8, 2015 Outline Classical results Radical of perfect number and the ABC-conjecture Idea of the proof Let σ(n) = d n d.
More informationSome new representation problems involving primes
A talk given at Hong Kong Univ. (May 3, 2013) and 2013 ECNU q-series Workshop (Shanghai, July 30) Some new representation problems involving primes Zhi-Wei Sun Nanjing University Nanjing 210093, P. R.
More informationOn a combinatorial method for counting smooth numbers in sets of integers
On a combinatorial method for counting smooth numbers in sets of integers Ernie Croot February 2, 2007 Abstract In this paper we develop a method for determining the number of integers without large prime
More informationRepresenting numbers as the sum of squares and powers in the ring Z n
Representing numbers as the sum of squares powers in the ring Z n Rob Burns arxiv:708.03930v2 [math.nt] 23 Sep 207 26th September 207 Abstract We examine the representation of numbers as the sum of two
More informationSQUARE 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
More informationAVERAGE RECIPROCALS OF THE ORDER OF a MODULO n
AVERAGE RECIPROCALS OF THE ORDER OF a MODULO n KIM, SUNGJIN Abstract Let a > be an integer Denote by l an the multiplicative order of a modulo integers n We prove that l = an Oa ep 2 + o log log, n,n,a=
More informationSolved and unsolved problems in elementary number theory
Solved and unsolved problems in elementary number theory Paul Pollack Athens/Atlanta Number Theory Seminar February 25, 2014 1 of 45 For a less slanted view of the subject, consider purchasing: 2 of 45
More informationMath 229: Introduction to Analytic Number Theory Čebyšev (and von Mangoldt and Stirling)
ath 9: Introduction to Analytic Number Theory Čebyšev (and von angoldt and Stirling) Before investigating ζ(s) and L(s, χ) as functions of a complex variable, we give another elementary approach to estimating
More informationMATH 152 Problem set 6 solutions
MATH 52 Problem set 6 solutions. Z[ 2] is a Euclidean domain (i.e. has a division algorithm): the idea is to approximate the quotient by an element in Z[ 2]. More precisely, let a+b 2, c+d 2 Z[ 2] (of
More informationThe Average Number of Divisors of the Euler Function
The Average Number of Divisors of the Euler Function Sungjin Kim Santa Monica College/Concordia University Irvine Department of Mathematics i707107@math.ucla.edu Dec 17, 2016 Euler Function, Carmichael
More informationSmol Results on the Möbius Function
Karen Ge August 3, 207 Introduction We will address how Möbius function relates to other arithmetic functions, multiplicative number theory, the primitive complex roots of unity, and the Riemann zeta function.
More informationSolutions of the problem of Erdös-Sierpiński:
arxiv:0707.2190v1 [math.nt] 15 Jul 2007 Solutions of the problem of Erdös-Sierpiński: σ(n) = σ(n+1) Lourdes Benito lourdes.benito@estudiante.uam.es Abstract For n 1.5 10 10, we have found a total number
More informationcongruences Shanghai, July 2013 Simple proofs of Ramanujan s partition congruences Michael D. Hirschhorn
s s m.hirschhorn@unsw.edu.au Let p(n) be the number of s of n. For example, p(4) = 5, since we can write 4 = 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1 and there are 5 such representations. It was known to Euler
More informationA CONSTRUCTION OF ARITHMETIC PROGRESSION-FREE SEQUENCES AND ITS ANALYSIS
A CONSTRUCTION OF ARITHMETIC PROGRESSION-FREE SEQUENCES AND ITS ANALYSIS BRIAN L MILLER & CHRIS MONICO TEXAS TECH UNIVERSITY Abstract We describe a particular greedy construction of an arithmetic progression-free
More informationOrder and chaos. Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA
Order and chaos Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA Perfect shuffles Suppose you take a deck of 52 cards, cut it in half, and perfectly shuffle it (with the bottom card staying
More informationarxiv:math.nt/ v1 5 Jan 2005
NEW TECHNIQUES FOR BOUNDS ON THE TOTAL NUMBER OF PRIME FACTORS OF AN ODD PERFECT NUMBER arxiv:math.nt/0501070 v1 5 Jan 2005 KEVIN G. HARE Abstract. Let σ(n) denote the sum of the positive divisors of n.
More informationIndustrial Strength Factorization. Lawren Smithline Cornell University
Industrial Strength Factorization Lawren Smithline Cornell University lawren@math.cornell.edu http://www.math.cornell.edu/~lawren Industrial Strength Factorization Given an integer N, determine the prime
More informationPrime Divisors of Palindromes
Prime Divisors of Palindromes William D. Banks Department of Mathematics, University of Missouri Columbia, MO 6511 USA bbanks@math.missouri.edu Igor E. Shparlinski Department of Computing, Macquarie University
More informationAN EXTENSION OF A THEOREM OF EULER. 1. Introduction
AN EXTENSION OF A THEOREM OF EULER NORIKO HIRATA-KOHNO, SHANTA LAISHRAM, T. N. SHOREY, AND R. TIJDEMAN Abstract. It is proved that equation (1 with 4 109 does not hold. The paper contains analogous result
More informationPseudoprime Statistics to 10 19
Pseudoprime Statistics to 10 19 Jens Kruse Andersen and Harvey Dubner CONTENTS 1. Introduction 2. Background Information 3. Results References A base-b pseudoprime (psp) is a composite N satisfying b N
More informationSignature: (In Ink) UNIVERSITY OF MANITOBA TEST 1 SOLUTIONS COURSE: MATH 2170 DATE & TIME: February 11, 2019, 16:30 17:15
PAGE: 1 of 7 I understand that cheating is a serious offence: Signature: (In Ink) PAGE: 2 of 7 1. Let a, b, m, be integers, m > 1. [1] (a) Define a b. Solution: a b iff for some d, ad = b. [1] (b) Define
More informationResults of modern sieve methods in prime number theory and more
Results of modern sieve methods in prime number theory and more Karin Halupczok (WWU Münster) EWM-Conference 2012, Universität Bielefeld, 12 November 2012 1 Basic ideas of sieve theory 2 Classical applications
More informationTWO REMARKS ON ITERATES OF EULER S TOTIENT FUNCTION
TWO REMARKS ON ITERATES OF EULER S TOTIENT FUNCTION PAUL POLLACK Abstract. Let ϕ k denote the kth iterate of Euler s ϕ-function. We study two questions connected with these iterates. First, we determine
More informationOn Integers for Which the Sum of Divisors is the Square of the Squarefree Core
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 15 (01), Article 1.7.5 On Integers for Which the Sum of Divisors is the Square of the Squarefree Core Kevin A. Broughan Department of Mathematics University
More informationOn Dris conjecture about odd perfect numbers
Notes on Number Theory and Discrete Mathematics Print ISSN 1310 513, Online ISSN 367 875 Vol. 4, 018, No. 1, 5 9 DOI: 10.7546/nntdm.018.4.1.5-9 On Dris conjecture about odd perfect numbers Paolo Starni
More informationCHAPTER 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
More informationProbability and Number Theory: an Overview of the Erdős-Kac Theorem
Probability and umber Theory: an Overview of the Erdős-Kac Theorem December 12, 2012 Probability and umber Theory: an Overview of the Erdős-Kac Theorem Probabilistic number theory? Pick n with n 10, 000,
More information198 VOLUME 46/47, NUMBER 3
LAWRENCE SOMER Abstract. Rotkiewicz has shown that there exist Fibonacci pseudoprimes having the forms p(p + 2), p(2p 1), and p(2p + 3), where all the terms in the products are odd primes. Assuming Dickson
More informationEquidivisible consecutive integers
& Equidivisible consecutive integers Ivo Düntsch Department of Computer Science Brock University St Catherines, Ontario, L2S 3A1, Canada duentsch@cosc.brocku.ca Roger B. Eggleton Department of Mathematics
More informationNumber Theory Marathon. Mario Ynocente Castro, National University of Engineering, Peru
Number Theory Marathon Mario Ynocente Castro, National University of Engineering, Peru 1 2 Chapter 1 Problems 1. (IMO 1975) Let f(n) denote the sum of the digits of n. Find f(f(f(4444 4444 ))). 2. Prove
More informationAn 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
More informationOn the largest prime factor of the Mersenne numbers
arxiv:0704.137v1 [math.nt] 10 Apr 007 On the largest prime factor of the Mersenne numbers Kevin Ford Department of Mathematics The University of Illinois at Urbana-Champaign Urbana Champaign, IL 61801,
More informationTHE FIRST FUNCTION AND ITS ITERATES
THE FIRST FUNCTION AND ITS ITERATES CARL POMERANCE For Ron Graham on his 80th birthday. Abstract. Let s(n) denote the sum of the positive divisors of n ecept for n iteself. Discussed since Pythagoras,
More informationp = 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
More informationOn prime factors of subset sums
On prime factors of subset sums by P. Erdös, A. Sárközy and C.L. Stewart * 1 Introduction For any set X let X denote its cardinality and for any integer n larger than one let ω(n) denote the number of
More informationOn 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
More information