On the number of co-prime-free sets. Neil J. Calkin and Andrew Granville *
|
|
- Annabella Norton
- 5 years ago
- Views:
Transcription
1 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 positive integers, that have that property. In so doing we answer some questions posed by Cameron and Erdös. 1. Introduction. In [CE] Cameron and Erdös investigated subsets of the positive integers with certain given properties P ; in particular, how large such sets can be, and how many there are. The properties P that they were interested in are monotone decreasing, that is, if S has property P, and T is a subset of S, then T has property P. Thus if S is a maimal set of positive integers with property P then one knows that there are 2 S such sets. In this paper we improve various estimates in [CE] for the number of sets satisfying certain properties P : In Theorem 3.5 of [CE], Cameron and Erdös showed that the number of sets of positive integers, in which any two elements have a common factor, lies between 2 [/2] and 2 [/2]. Here we improve this to Theorem 1. The number of sets of integers, with any two elements having a common factor, is 1.1) 2 [/2] + 2 [/2] N + O 2 [/2] N ep C log 2 log log )), * The second author is supported, in part, by the National Science Foundation grant number DMS )
2 for some absolute constant C > 0, where N, which will be defined in the proof, satisfies 1.2) N = e γ + o1)) log log, and γ is the Euler-Mascheroni constant. In Theorem 3.3 of [CE], Cameron and Erdös showed that the number of sets of positive integers, in which any two elements are coprime, lies between 2 π) e 1/2+o1)) and 2 π) e 2+o1)). Here we improve this to Theorem 2. The number of sets of integers, with every pair of elements coprime, is 1.3) 2 π) e {1+Olog log /log )}. In Section 4.3 of [CE], Cameron and Erdös conjectured that there are cs) +o) sets of integers, with sum of reciprocals bounded by s, for some positive constant cs). We prove a quantitatative form of this conjecture here: Theorem 3. The number, ν), of sets {a 1, a 2,... a t } of positive integers, with i 1/a i s, is 1.4) cs) e O3/4), where cs) = and fs) is defined by 1.5) s = fs) du u1 + e u ). 1 + e fs)), Cameron and Erdo s observed that cs) 2 1 e s, for all s. We can provide some rather more accurate estimates based on an analysis of 1.4) and 1.5): As s 0, we have 1.6) cs) = 1 + s log 1/s) + log log 1/s) + O1)) ; as s, we have 1.7) cs) = 2 Ce 2s + Oe 4s ),
3 for some constant C On the number of co-prime-free sets 3 2. Sets where any two elements have a common factor. Proof of Theorem 1: Clearly there are 2 [/2] such sets that contain only even numbers. We now count such sets that contain at least one odd number: Let k be the largest integer for which q = p 1 p 2... p k where p j is the jth smallest odd prime); by the Prime Number Theorem, k log /log log. Define R = {n : n is divisible by 2p j for some j k}. Clearly any set of the form S {q}, where S is a subset of R, has the property that any two elements have a common factor. The number of such subsets S is 2 R, and R = [/2] N where N is the number of integers 2m that are coprime to q. From the combinatorial sieve we obtain N φq) q 2, and then Mertens Theoremplies 1.2). Our proof of the rest of Theorem 1 is based on the ideas of Pomerance given in Cameron and Erdös [CE]: We start by ordering the odd numbers as q = m 1, m 2,... so that φm 1 ) m 1 φm 2) m 2 φm 3) m Define f i ) to be the number of sets of integers, that contain but not +1, +2,..., and for which every pair of integers in the set have a common factor. If φ) 2/3 then #{n : n, ) > 1} p p p log ) p = log p 1 ) mi log 3/2), φ ) so that f i ) 2 log 3/2), which is part of the error tern 1.1).
4 4 Neil J. Calkin and Andrew Granville Define A m ) to be the number of even integers, that are coprime to m, and c i = A mi ). Clearly f i ) 2 i 1+[/2] c i, and so, in order to complete the proof of Theorem 1, we must show that 2.1) c i ) i N log 2 log log, for all i 2 for which φ) 2/3. Suppose that m = pr m = p r are odd, squarefree numbers, with p < 1/3. Noting that A r ) = A m ) + A r /p), we see that 2.2) A m ) A m ) = A r /p) \ A r /p ) 1 p 1 ) p log log p 2 log log. Thus, given any squarefree integer r, r q, we form a sequence r = r 0, r 1,... r j = q, as follows: If r i = p 1 p 2... p h for some h < k, then then let r i+1 = r i p h+1. Otherwise we construct r i+1 by dividing r i by its largest prime factor, and multiplying it by the smallest odd prime that does not yet divide it. Now as A q ) = N, and as the prime factors of q are all log, we find that, if i 2 then c i N /log 2 log log by 2.2); which implies 2.1) for i /log 2 log log. So we are left with those i for which φ) deal with those i 100/log log : 2/3 and i /log 2 log log. We first If A is any set of < i odd integers then φ ) min n, n odd n A φn) n. So we choose A to be the set of those [/log 3 ] odd integers, with the most distinct prime factors. Hardy and Ramanujan [HR] showed that there eists an absolute constant c such that the number of integers with eactly k distinct prime factors is log log log + c) k 1 ; k 1)!
5 On the number of co-prime-free sets 5 therefore the number of integers with at least 10log log distinct prime factors is /log 10. Thus, if n A then n has 10log log distinct prime factors, and so φn) n p log log ) ) p 1 log log log by Mertens Theorem. Therefore, by the combinatorial sieve, which implies 2.1) in this range of i. c i φ) log log log, Finally we come to those i in the range 100/log log i /2, with φ) 2/3: An immediate consequence of Proposition 4 of [PS] is that i < 3 / 20 φ) 1 φm ) i) 9 20 φ). On the other hand, by the combinatorial sieve, { } 1 c i 2 + o1) φ) > { } o1) i; and thus and so 2.1) is satisfied. c i i i log log, 3. Sets where any two elements are coprime. Proof of Theorem 2: Let t = π ). For the upper bound, note that the number of composite elements of each set is t, as these elements must all have distinct prime factors. All other elements are prime, and so the number of such sets is 2 π) t ) [] i i=0 π) t 2 t! 2 π) e ) t, t
6 6 Neil J. Calkin and Andrew Granville which gives 1.3) by the Prime Number Theorem. The proof here is the same as in Theorem 3.3 of [CE], ecept that they made a computational error in the final step.) and let To obtain the lower bound, we shall construct 1.3) such sets. Let ) log log k = t 1 log < q1 < q 2 <... < q k be the k smallest primes larger than. Note that, using the Prime Number Theoren the form π) = log 1 + O1/log )) we have 3.1) q j = 1/2 1 + j/t + O1/log )) and so q k < 2. We construct our sets as follows: Each prime in the interval 2, ] is in our set or not as desired, giving 2 π) π2 ) different options. We may put any number of the form p k q k in the set, where p k is a prime less than /q k giving π/q k ) choices). Then any p k 1 q k 1 where p k 1 is a prime /q k 1 giving π/q k 1 ) 1 choices). We continue in this fashion, taking, in general, any p k j q k j, where p k j is a prime /q k j, not already used as some p k i, giving us π/q k j ) j choices), for j = 0, 1,..., k 1. Thus the number of different sets constructed is k { ) π Now, by 3.1), ) π q i 2 π)+o /log ) i=0 1/2 k i) = π 1 + i/t = t { i/t + O q i } k i). ))) O k i) log ) 1 log log 1 + log log + i t },
7 On the number of co-prime-free sets 7 using the Prime Number Theorem again, )) log log = t log + i/t) i/t + O log Therefore, the number of sets is at least t log. which gives 1.3). 2 π)+o /log ) ) k t, log Remark: With some care it is possible to replace the log log in 1.3) with log log log ) 2, but we are currently unable to do better. 4. Sets whose sum of reciprocals is bounded. Proof of Theorem 3: Let y = [ 1/4 ], z = [ 1/2 ], and j = j/z for y j z. A given set of positive integers, whose sum of reciprocals is bounded by s, has, say, b j integers in the interval j, j+1 ] for y j < z with 0 b j /z + 1), and so satisfies 4.1) z 1 j=y b j j+1 s. So, if the b j are fied with these values then the number of sets of integers, with precisely b j integers from the interval j, j+1 ], is 4.2) 2 y/z y j<z ) [/z] + 1. So define α j := b j /[/z] + 1) for each j. Clearly 0 α j 1 for each j, and, by 4.1), they must satisfy 4.3) z 1 j=y α j j s 1 + O b j )) 1. y
8 8 Neil J. Calkin and Andrew Granville Moreover, by Stirling s formula, ) [/z] + 1 b j ep Olog ) ) z α jlog α j + 1 α j )log 1 α j )). Noting that there are no more than /z + 1) z choices for the b j, we thus see that ν) ep O 3/4 ) z min 0 α j 1,y j<z 4.3) holds y j<z α j log α j + 1 α j )log 1 α j )). By the method of Lagrange multipliers we find that the minimum occurs when each α j = 1/ 1 + e A/j) for some constant A > 0. Now z 1 j=y α j j = = z y A/z dt 1 t1 + e A/t ) + O ye + 1 A/y du u1 + e u ) + O ze A/z ) 1 ye A/y + 1 A/y)e A/y + 1 ze A/z ). To obtain equality in 4.3), we need to select A = 1/2 fs) + O 1/4 ). Thus y j<z α j log α j + 1 α j )log 1 α j ) = = z y 1 + e A/u) + By the substitution t = A/u, and the fact that d dt log 1 + e t ) t y j<z log A u 1 + e A/u) 1 + e A/j) + A j ) = 1 t 2 log 1 + e t ) + du + O1). t ) 1 + e t, e A/j ) this last line is zlog 1 + e A/z ) + O1), and so 1.4) is an upper bound for ν).
9 On the number of co-prime-free sets 9 To obtain a lower bound for ν) note that if we select integers b j such that 4.4) z 1 j=y b j j s, then the sum of the reciprocals of any set consisting of b j j, j+1 ), for each y j < z, is s. Clearly the number of such sets is 4.5) y j<z ) [/z]. b j integers from the interval Now the idea is to select each b j = /z 1 + e A/j) + O1), for some constant A > 0, so that 4.4) is satisfied; this can be done by the choice A = 1/2 fs) + O 1/4 ) the proof being almost identical to that for the upper bound). Now, when we estimate 4.5) with this value for A, we proceed as in the upper bound, and show that 4.5) is at least 1.4). Remarks: Cameron and Erdo s observed that any set of integers taken from [/e s, ] has sum of reciprocals s, and thus cs) 2 1 e s. We will derive 1.6) and 1.7): If is large then du u1 + e u ) = 1 { e 1 + O )} 1, which implies 1.6). On the other hand, 1.5) gives that, for fs) < 1 that is, s large ), 4.6) s = C log 1/fs)) fs) where C 1 = 1 du u1 + e u ) e u 1 ) du 2 u, e u 1 ) du 2 u. Therefore fs) e 2s, and so the last tern 4.6) is fs) e 2s. Thus fs) = Ce 2s +Oe 4s ), where C = e 2C 1 which can be computed eplicitly), which implies 1.7).
10 10 Neil J. Calkin and Andrew Granville References [CE] CAMERON, P.J. & ERDÖS, P., On the number of sets of integers with various properties, Number Theory ed. R.A. Mollin), de Gruyter, New York, 1990), [HR] [PS] HARDY, G.H. & RAMANUJAN, S,, The normal number of prime factors of a number n, Quarterly J. Math., ), POMERANCE, C. & SELFRIDGE, J.L., Proof of D.J. Newman s coprime mapping conjecture, Mathematika, ), Neil J. Calkin, School of Mathematics, Georgia Institute of Technology, Altanta, GA 30332, USA. Andrew J. Granville, Department of Mathematics, University of Georgia, GA 30602, USA.
Euler 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 informationMath 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
More informationOn the average number of divisors of the Euler function
On the average number of divisors of the Euler function Florian Luca Instituto de Matemáticas Universidad Nacional Autonoma de Méico CP 58089, Morelia, Michoacán, Méico fluca@matmorunamm Carl Pomerance
More informationCompositions with the Euler and Carmichael Functions
Compositions with the Euler and Carmichael Functions William D. Banks Department of Mathematics University of Missouri Columbia, MO 652, USA bbanks@math.missouri.edu Florian Luca Instituto de Matemáticas
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 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 informationTHE CONVEX HULL OF THE PRIME NUMBER GRAPH
THE CONVEX HULL OF THE PRIME NUMBER GRAPH NATHAN MCNEW Abstract Let p n denote the n-th prime number, and consider the prime number graph, the collection of points n, p n in the plane Pomerance uses the
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 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 informationON A CLASS OF APERIODIC SUM-FREE SETS NEIL J. CALKIN PAUL ERD OS. that is that there are innitely many n not in S which are not a sum of two elements
ON A CLASS OF APERIODIC SUM-FREE SETS NEIL J. CALKIN PAUL ERD OS Abstract. We show that certain natural aperiodic sum-free sets are incomplete, that is that there are innitely many n not in S which are
More informationRepeated Values of Euler s Function. a talk by Paul Kinlaw on joint work with Jonathan Bayless
Repeated Values of Euler s Function a talk by Paul Kinlaw on joint work with Jonathan Bayless Oct 4 205 Problem of Erdős: Consider solutions of the equations ϕ(n) = ϕ(n + k), σ(n) = σ(n + k) for fixed
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 informationPALINDROMIC SUMS OF PROPER DIVISORS. Paul Pollack Department of Mathematics, University of Georgia, Athens, Georgia
#A3 INTEGERS 5A (205) PALINDROMIC SUMS OF PROPER DIVISORS Paul Pollack Department of Mathematics, University of Georgia, Athens, Georgia pollack@math.uga.edu Received: 0/29/3, Accepted: 6/8/4, Published:
More informationarxiv:math/ v1 [math.nt] 17 Apr 2006
THE CARMICHAEL NUMBERS UP TO 10 18 arxiv:math/0604376v1 [math.nt] 17 Apr 2006 RICHARD G.E. PINCH Abstract. We extend our previous computations to show that there are 1401644 Carmichael numbers up to 10
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 information10 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
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 informationOn the maximal exponent of the prime power divisor of integers
Acta Univ. Sapientiae, Mathematica, 7, 205 27 34 DOI: 0.55/ausm-205-0003 On the maimal eponent of the prime power divisor of integers Imre Kátai Faculty of Informatics Eötvös Loránd University Budapest,
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 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 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 informationProducts of ratios of consecutive integers
Products of ratios of consecutive integers Régis de la Bretèche, Carl Pomerance & Gérald Tenenbaum 27/1/23, 9h26 For Jean-Louis Nicolas, on his sixtieth birthday 1. Introduction Let {ε n } 1 n
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 informationIntermediate Math Circles February 14, 2018 Contest Prep: Number Theory
Intermediate Math Circles February 14, 2018 Contest Prep: Number Theory Part 1: Prime Factorization A prime number is an integer greater than 1 whose only positive divisors are 1 and itself. An integer
More information= 1 2 x (x 1) + 1 {x} (1 {x}). [t] dt = 1 x (x 1) + O (1), [t] dt = 1 2 x2 + O (x), (where the error is not now zero when x is an integer.
Problem Sheet,. i) Draw the graphs for [] and {}. ii) Show that for α R, α+ α [t] dt = α and α+ α {t} dt =. Hint Split these integrals at the integer which must lie in any interval of length, such as [α,
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 informationarxiv: v2 [math.nt] 28 Jun 2012
ON THE DIFFERENCES BETWEEN CONSECUTIVE PRIME NUMBERS I D. A. GOLDSTON AND A. H. LEDOAN arxiv:.3380v [math.nt] 8 Jun 0 Abstract. We show by an inclusion-eclusion argument that the prime k-tuple conjecture
More informationarxiv: v1 [math.nt] 22 Jun 2017
lexander Kalmynin arxiv:1706.07343v1 [math.nt] 22 Jun 2017 Nová-Carmichael numbers and shifted primes without large prime factors bstract. We prove some new lower bounds for the counting function N C (x)
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 informationON A PROBLEM OF ARNOLD: THE AVERAGE MULTIPLICATIVE ORDER OF A GIVEN INTEGER
ON A PROBLEM OF ARNOLD: THE AVERAGE MULTIPLICATIVE ORDER OF A GIVEN INTEGER Abstract. For g, n coprime integers, let l g n denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold,
More informationProduct of integers in an interval, modulo squares
Product of integers in an interval, modulo squares Andrew Granville Department of Mathematics, University of Georgia Athens, GA 30602-7403 and J.L. Selfridge Department of Mathematics,
More informationDisproof 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
More informationSummary Slides for MATH 342 June 25, 2018
Summary Slides for MATH 342 June 25, 2018 Summary slides based on Elementary Number Theory and its applications by Kenneth Rosen and The Theory of Numbers by Ivan Niven, Herbert Zuckerman, and Hugh Montgomery.
More informationCHAPTER 1 REAL NUMBERS KEY POINTS
CHAPTER 1 REAL NUMBERS 1. Euclid s division lemma : KEY POINTS For given positive integers a and b there exist unique whole numbers q and r satisfying the relation a = bq + r, 0 r < b. 2. Euclid s division
More informationDense Admissible Sets
Dense Admissible Sets Daniel M. Gordon and Gene Rodemich Center for Communications Research 4320 Westerra Court San Diego, CA 92121 {gordon,gene}@ccrwest.org Abstract. Call a set of integers {b 1, b 2,...,
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 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 informationMath 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
More informationPrimality 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
More informationVarious new observations about primes
A talk given at Univ. of Illinois at Urbana-Champaign (August 28, 2012) and the 6th National Number Theory Confer. (October 20, 2012) and the China-Korea Number Theory Seminar (October 27, 2012) Various
More informationON THE SET OF REDUCED φ-partitions OF A POSITIVE INTEGER
ON THE SET OF REDUCED φ-partitions OF A POSITIVE INTEGER Jun Wang Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P.R. China Xin Wang Department of Applied Mathematics,
More informationSOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM
Acta Arith. 183(018), no. 4, 339 36. SOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM YU-CHEN SUN AND ZHI-WEI SUN Abstract. Lagrange s four squares theorem is a classical theorem in number theory. Recently,
More informationProbabilistic Aspects of the Integer-Polynomial Analogy
Probabilistic Aspects of the Integer-Polynomial Analogy Kent E. Morrison Department of Mathematics California Polytechnic State University San Luis Obispo, CA 93407 kmorriso@calpoly.edu Zhou Dong Department
More informationON THE EQUATION τ(λ(n)) = ω(n) + k
ON THE EQUATION τ(λ(n)) = ω(n) + k A. GLIBICHUK, F. LUCA AND F. PAPPALARDI Abstract. We investigate some properties of the positive integers n that satisfy the equation τ(λ(n)) = ω(n)+k providing a complete
More informationDIVISIBILITY PROPERTIES OF THE 5-REGULAR AND 13-REGULAR PARTITION FUNCTIONS
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (008), #A60 DIVISIBILITY PROPERTIES OF THE 5-REGULAR AND 13-REGULAR PARTITION FUNCTIONS Neil Calkin Department of Mathematical Sciences, Clemson
More informationON THE ORDER OF UNIMODULAR MATRICES MODULO INTEGERS PRELIMINARY VERSION
ON THE ORDER OF UNIMODULAR MATRICES MODULO INTEGERS PRELIMINARY VERSION. Introduction Given an integer b and a prime p such that p b, let ord p (b) be the multiplicative order of b modulo p. In other words,
More informationA Guide to Arithmetic
A Guide to Arithmetic Robin Chapman August 5, 1994 These notes give a very brief resumé of my number theory course. Proofs and examples are omitted. Any suggestions for improvements will be gratefully
More informationGENERALIZATIONS OF SOME ZERO-SUM THEOREMS. Sukumar Das Adhikari Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad , INDIA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A52 GENERALIZATIONS OF SOME ZERO-SUM THEOREMS Sukumar Das Adhikari Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad
More informationPRIME DIVISORS ARE POISSON DISTRIBUTED
February, 200 6: WSPC/INSTRUCTION FILE 000 International Journal of Number Theory Vol., No. (200) 8 c World Scientific Publishing Company PRIME DIVISORS ARE POISSON DISTRIBUTED ANDREW GRANVILLE Départment
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 information#A42 INTEGERS 10 (2010), ON THE ITERATION OF A FUNCTION RELATED TO EULER S
#A42 INTEGERS 10 (2010), 497-515 ON THE ITERATION OF A FUNCTION RELATED TO EULER S φ-function Joshua Harrington Department of Mathematics, University of South Carolina, Columbia, SC 29208 jh3293@yahoo.com
More informationPrimitive roots in algebraic number fields
Primitive roots in algebraic number fields Research Thesis Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Joseph Cohen Submitted to the Senate of the Technion
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 informationAsymptotic formulae for the number of repeating prime sequences less than N
otes on umber Theory and Discrete Mathematics Print ISS 30 532, Online ISS 2367 8275 Vol. 22, 206, o. 4, 29 40 Asymptotic formulae for the number of repeating prime sequences less than Christopher L. Garvie
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 informationContradictions in some primes conjectures
Contradictions in some primes conjectures VICTOR VOLFSON ABSTRACT. This paper demonstrates that from the Cramer's, Hardy-Littlewood's and Bateman- Horn's conjectures (suggest that the probability of a
More information1 i<j k (g ih j g j h i ) 0.
CONSECUTIVE PRIMES IN TUPLES WILLIAM D. BANKS, TRISTAN FREIBERG, AND CAROLINE L. TURNAGE-BUTTERBAUGH Abstract. In a stunning new advance towards the Prime k-tuple Conjecture, Maynard and Tao have shown
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 informationDIVISOR-SUM FIBERS PAUL POLLACK, CARL POMERANCE, AND LOLA THOMPSON
DIVISOR-SUM FIBERS PAUL POLLACK, CARL POMERANCE, AND LOLA THOMPSON Abstract. Let s( ) denote the sum-of-proper-divisors function, that is, s(n) = d n, d
More informationClimbing an Infinite Ladder
Section 5.1 Section Summary Mathematical Induction Examples of Proof by Mathematical Induction Mistaken Proofs by Mathematical Induction Guidelines for Proofs by Mathematical Induction Climbing an Infinite
More informationOn the order of unimodular matrices modulo integers
ACTA ARITHMETICA 0.2 (2003) On the order of unimodular matrices modulo integers by Pär Kurlberg (Gothenburg). Introduction. Given an integer b and a prime p such that p b, let ord p (b) be the multiplicative
More informationAliquot sums of Fibonacci numbers
Aliquot sums of Fibonacci numbers Florian Luca Instituto de Matemáticas Universidad Nacional Autónoma de Méico C.P. 58089, Morelia, Michoacán, Méico fluca@matmor.unam.m Pantelimon Stănică Naval Postgraduate
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 informationON THE NUMBER OF SUBSEQUENCES WITH GIVEN SUM OF SEQUENCES IN FINITE ABELIAN p-groups
ON THE NUMBER OF SUBSEQUENCES WITH GIVEN SUM OF SEQUENCES IN FINITE ABELIAN p-groups WEIDONG GAO AND ALFRED GEROLDINGER Abstract. Let G be an additive finite abelian p-group. For a given (long) sequence
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 informationPermutation groups, simple groups, and sieve methods
Permutation groups, simple groups, and sieve methods D.R. Heath-Brown Mathematical Institute 24-29, St. Giles Oford, OX1 3LB England Cheryl E. Praeger School of Mathematics and Statistics University of
More information2 More on Congruences
2 More on Congruences 2.1 Fermat s Theorem and Euler s Theorem definition 2.1 Let m be a positive integer. A set S = {x 0,x 1,,x m 1 x i Z} is called a complete residue system if x i x j (mod m) whenever
More information= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2
8. p-adic numbers 8.1. Motivation: Solving x 2 a (mod p n ). Take an odd prime p, and ( an) integer a coprime to p. Then, as we know, x 2 a (mod p) has a solution x Z iff = 1. In this case we can suppose
More information1981] 209 Let A have property P 2 then [7(n) is the number of primes not exceeding r.] (1) Tr(n) + c l n 2/3 (log n) -2 < max k < ir(n) + c 2 n 2/3 (l
208 ~ A ~ 9 ' Note that, by (4.4) and (4.5), (7.6) holds for all nonnegatíve p. Substituting from (7.6) in (6.1) and (6.2) and evaluating coefficients of xm, we obtain the following two identities. (p
More informationDiscrete Mathematics with Applications MATH236
Discrete Mathematics with Applications MATH236 Dr. Hung P. Tong-Viet School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg Campus Semester 1, 2013 Tong-Viet
More informationChapter 5. Modular arithmetic. 5.1 The modular ring
Chapter 5 Modular arithmetic 5.1 The modular ring Definition 5.1. Suppose n N and x, y Z. Then we say that x, y are equivalent modulo n, and we write x y mod n if n x y. It is evident that equivalence
More informationOn Legendre s formula and distribution of prime numbers
On Legendre s formula and distribution of prime numbers To Hiroshi, Akiko, and Yuko Kazuo AKIYAMA Tatsuhiko NAGASAWA 1 Abstract The conclusion in this paper is based on the several idea obtained in recherché
More informationMath 110 HW 3 solutions
Math 0 HW 3 solutions May 8, 203. For any positive real number r, prove that x r = O(e x ) as functions of x. Suppose r
More informationON THE UNIFORM DISTRIBUTION OF CERTAIN SEQUENCES INVOLVING THE EULER TOTIENT FUNCTION AND THE SUM OF DIVISORS FUNCTION
Annales Univ. Sci. Budapest., Sect. Comp. 44 (205) 79 9 ON THE UNIFORM DISTRIBUTION OF CERTAIN SEQUENCES INVOLVING THE EULER TOTIENT FUNCTION AND THE SUM OF DIVISORS FUNCTION Jean-Marie De Koninck (Québec,
More informationARITHMETIC OF THE 13-REGULAR PARTITION FUNCTION MODULO 3
ARITHMETIC OF THE 13-REGULAR PARTITION FUNCTION MODULO 3 JOHN J WEBB Abstract. Let b 13 n) denote the number of 13-regular partitions of n. We study in this paper the behavior of b 13 n) modulo 3 where
More informationGENERATING RANDOM FACTORED GAUSSIAN INTEGERS, EASILY
GENERATING RANDOM FACTORED GAUSSIAN INTEGERS, EASILY NOAH LEBOWITZ-LOCKARD AND CARL POMERANCE Abstract. We present arandom) polynomial-time algorithm to generate a random Gaussian integer with the uniform
More informationProof of Infinite Number of Triplet Primes. Stephen Marshall. 28 May Abstract
Proof of Infinite Number of Triplet Primes Stephen Marshall 28 May 2014 Abstract This paper presents a complete and exhaustive proof that an Infinite Number of Triplet Primes exist. The approach to this
More informationWe only consider r finite. P. Erdős, On integers of the form 2 k +p and some related problems, Summa Brasil. Math. 2 (1950),
U niv e rsit y o f S o u th C ar o l ina e lf i las e ta b y M i cha A covering of the integers is a system of congruences x a j (mod m j ), j = 1, 2,..., r, with a j and m j integral and with m j 1, such
More informationarxiv: v1 [math.nt] 30 Oct 2017
SHIFTS OF THE PRIME DIVISOR FUNCTION OF ALLADI AND ERDŐS arxiv:1710.10875v1 [math.nt] 30 Oct 2017 SNEHAL SHEKATKAR AND TIAN AN WONG Abstract. We introduce a variation on the prime divisor function B(n)
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 informationPrimality 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
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 informationNecessary and Sufficient Conditions for the Central Norm to Equal 2 h in the Simple Continued Fraction Expansion of 2 h c for Any Odd Non-Square c > 1
Necessary and Sufficient Conditions for the Central Norm to Equal 2 h in the Simple Continued Fraction Expansion of 2 h c for Any Odd Non-Square c > 1 R.A. Mollin Abstract We look at the simple continued
More informationAll variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points.
Math 152, Problem Set 2 solutions (2018-01-24) All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points. 1. Let us look at the following equation: x 5 1
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 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 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 informationCHARACTER-FREE APPROACH TO PROGRESSION-FREE SETS
CHARACTR-FR APPROACH TO PROGRSSION-FR STS VSVOLOD F. LV Abstract. We present an elementary combinatorial argument showing that the density of a progression-free set in a finite r-dimensional vector space
More informationON THE SUM OF DIVISORS FUNCTION
Annales Univ. Sci. Budapest., Sect. Comp. 40 2013) 129 134 ON THE SUM OF DIVISORS FUNCTION N.L. Bassily Cairo, Egypt) Dedicated to Professors Zoltán Daróczy and Imre Kátai on their 75th anniversary Communicated
More informationNUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:
NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two operations defined on them, addition and multiplication,
More informationEXPLICIT CONSTANTS IN AVERAGES INVOLVING THE MULTIPLICATIVE ORDER
EXPLICIT CONSTANTS IN AVERAGES INVOLVING THE MULTIPLICATIVE ORDER KIM, SUNGJIN DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, LOS ANGELES MATH SCIENCE BUILDING 667A E-MAIL: 70707@GMAILCOM Abstract
More informationHOMEWORK 4 SOLUTIONS TO SELECTED PROBLEMS
HOMEWORK 4 SOLUTIONS TO SELECTED PROBLEMS 1. Chapter 3, Problem 18 (Graded) Let H and K be subgroups of G. Then e, the identity, must be in H and K, so it must be in H K. Thus, H K is nonempty, so we can
More informationPrimality 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
More informationCONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ω(q) AND ν(q)
CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE Abstract. Recently, Andrews, Dixit and Yee introduced partition
More informationELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS
Bull Aust Math Soc 81 (2010), 58 63 doi:101017/s0004972709000525 ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS MICHAEL D HIRSCHHORN and JAMES A SELLERS (Received 11 February 2009) Abstract
More informationMath 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
More informationHomework #2 solutions Due: June 15, 2012
All of the following exercises are based on the material in the handout on integers found on the class website. 1. Find d = gcd(475, 385) and express it as a linear combination of 475 and 385. That is
More informationDense 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 informationMath 109 HW 9 Solutions
Math 109 HW 9 Solutions Problems IV 18. Solve the linear diophantine equation 6m + 10n + 15p = 1 Solution: Let y = 10n + 15p. Since (10, 15) is 5, we must have that y = 5x for some integer x, and (as we
More informationC-COLOR COMPOSITIONS AND PALINDROMES
CAROLINE SHAPCOTT Abstract. An unepected relationship is demonstrated between n-color compositions compositions for which a part of size n can take on n colors) and part-products of ordinary compositions.
More information