1 Sieving an interval
|
|
- Sharleen Cooper
- 6 years ago
- Views:
Transcription
1 Math 05 notes, wee 9 C. Pomerance Sieving an interval It is interesting that our sieving apparatus that we have developed is robust enough to handle some problems on the distribution of prime numbers that would otherwise be unattacable. For example, we saw that if then A = {integers n : n (0,x]}, P = the set of primes, y = x /(0loglogx), As a corollary we have S(A,P,y) x logy xloglogx. logx π(x) xloglogx. logx Now say we mae a very small change. We alter the sequence A to A = {integers n : n (z,z +x]}, and we leave everything else unchanged. What allowed us to estimate S(A, P, y) was the estimate A d = x = = d x d +O(). a A d a What about A d? This is A d = a A d a x+z z = d d = x+z d z d +O() = x d +O(). That is, we have essentially the same estimate for A d as we had for A d. So, exactly the same calculation as before gives us And we have the corollary that S(A,P,y) x logy xloglogx. logx π(z +x) π(z) xloglogx. logx All of our sieve results wor in this more general way, allowing us to sift an interval.
2 2 The prime convexity conjecture In a paper from 923, Hardy and Littlewood laid out a great many conjectures on the distribution of prime numbers, giving them an analytic cast. For example, instead of just conjecturing that there are infinitely many twin primes, they conjectured that π 2 (x) 2C 2 x 2 dt (logt) 2, where C 2 is the twin prime constant discussed earlier. They had similar strong conjectures for prime -tuples, and they also had a strong version of Goldbach s conjecture (that even numbers starting at 4 are the sum of two primes). They also presented a new conjecture, now nown as the prime convexity conjecture: If x,z 2, then π(z +x) π(z)+π(x). That is, the champion (or co-champion) interval of length x for having prime numbers is the initial interval (0,x]. The interval (z,z+x] has π(z+x) π(z) primes, and the prime convexity conjecture asserts that this count cannot exceed π(x), the number of primes in (0, x]. It came as somewhat of a shoc in 973 when Hensley and Richards showed that the prime convexity conjecture is incompatible with the prime -tuples conjecture. Here is their idea. Suppose that {a,a 2,...,a }, 0 < a < a 2 < < a is admissible. Recall that this means that for every prime p, the a i s do not contain a complete residue system modulo p. The prime -tuples conjecture then asserts that there are infinitely many integers n with n+a,n+a 2,...,n+a all prime. Each time this happens we have π(n+a ) π(n). What Hensley and Richards found was a large admissible set with a < p, where p denotes the -th prime. The prime convexity conjecture with z = n and x = a would assert that π(z +x) π(z) π(x) = π(a ) π(p ) =, a contradiction. So, which conjecture is to be believed? Most people seem to put their faith (if you want to call it that) in the prime -tuples conjecture. The admissible set that Hensley and Richards came up with is quite concrete, though fairly large. To disprove the prime convexity conjecture, all one would need to do is to find some one integer n where all of n+a i are prime. However, this is not such an easy tas, since the admissible set is fairly large. People have also looed at trying to reduce the size of the admissible set in the proof, and there has been some success there. I believe it is the case still that none of the conjectures in the original Hardy and Littlewood paper have been settled. However, we do now for sure that at least one conjecture in the paper is false! 2
3 3 Periodic functions We have seen that a Dirichlet character χ mod is a periodic function with period that is also completely multiplicative. Let us loo now more generally at the set P of all arithmetic functions f : Z C which are periodic with period. It is clear that P is closed under addition of functions and under scalar multiplication (with the field of scalars being C). It is a complex vector space. Since the values f(),f(2),...,f() determine f and may be arbitrarily prescribed, it is clear that P has dimension. In particular, it is naturally isomorphic to C, the vector space of ordered -tuples of complex numbers. We loo now for a convenient basis for the vector space P. We of course have the natural one corresponding to the standard basis of C. But we would lie a basis that more overtly reflexts the periodicity modulo. Mathematics has long used the trig functions sin and cos to model periodic functions, and we can try the same thing. First, some convenient notation. It is cumbersome to write out e 2πiθ all the time, so we abbreviate this as e(θ). So, in this notation there is a little bit of ambiguity between the number e, and now the function e. To lessen this ambiguity, let us also define e (θ) = e(θ/) = e 2πiθ/. Note that e (θ ) = e (θ 2 ) if and only if θ θ 2 is an integer divisible by. For j = 0,,...,, let f j (n) = e (jn) = e 2πijn/. Then f j is a periodic function with period, so it is in P. We show that f 0,f,...,f form a basis for P and we do this by showing that every member g P can be written as a linear combination of f 0,f,...,f. That is, we wish to show that there are complex numbers c 0,c,...,c with g = c 0 f 0 +c f + +c f. In fact, there is a neat formula for these coefficients c j : Let s chec it out: c j = c j f j (n) = e (jn) g(m)f m ( j) = g(m)e ( jm) = g(m)e ( jm). g(m) e (j(n m)), since e (θ )e ( θ 2 ) = e (θ θ 2 ). The inner sum here can be evaluated, in fact it is a sum of a geometric progression. If n m (mod ), then e (j(n m)) = and the inner sum is, while otherwise, the inner sum is 0. Hence the total expression pulls out that value of g(m) where m n (mod ), and gives it weight =. So the above sum is g(n) and we indeed have g = c 0 f 0 +c f + +c f. 3
4 4 The Fourier coefficients of a Dirichlet character If we have a Dirichlet character mod, we recognize it as a periodic function mod that has some extra properties. Forgetting for a moment those extra properties, we can use the approach of the last section to write χ as a linear combination of the functions e (jn) for j = 0,,...,. As we saw, the coefficients c j are given by the expression There is a special notation for the sum here: Let G(j,χ) = χ(m)e ( jm). χ(m)e (jm). This is a Gauss sum, so the letter G is used. From what we did in the previous unit, we have χ(n) = G( j,χ)e (jn), n Z. () We now try and figure out some properties of Gauss sums. By trying to compute a few examples we notice that sometimes G(j,χ) = 0 and sometimes not. For some characters χ this seems to depend on whether j is coprime to, so G(j,χ) = 0 if and only if gcd(j,) >. And for some other characters this does not seem to be the case. To mae a longer story shorter, we cut to the chase. There are two types of Dirichlet characters: primitive and imprimitive. The definition of imprimitive is easier, the definition of primitive then being not imprimitive. Suppose that ψ is a character mod d, we have d, d <, and χ is the principal character mod. Then ψχ is a character mod and it is imprimitive. So, a primitive character mod is one that cannot be induced in this way by a character of smaller modulus. If is a prime, then every non-principal character mod is primitive, since has only the properdivisor. Herearethe4realcharactersmod5givenbytheirvaluesat,2,4,7,8,,3, 4:,,,,,,,,,,,,,,,,,,,,,,,,,,,,. Can you tell which are primitive and which are imprimitive? The ey is to loo at χ(7) and χ(), the 4th and 6th entries. Since 7 is mod 3 and a generator mod 5, if the value here is 4
5 , then we see that chi is induced from either a character mod or one mod 3, so the first two are not primitive. Since is mod 5 and a generator mod 3, we see that the first and third are not primitive. Only the 4th line is from a primitive character. Here is a criterion for being imprimitive: there is a divisor d with d < such that for each a (mod d) and gcd(a,) = we have χ(a) =. The proof of this is fairly easy; one shows that if u v (mod d) and gcd(uv,) = then χ(u) = χ(v), and so we are on the way to defining a character mod d that induces χ. Proposition. Suppose that χ is a character mod and gcd(n,) >. If G(n,χ) 0, then χ is not primitive. Proof. Let d = /gcd(n,) so that d and d <. Suppose that a is an integer coprime to and satisfying a (mod d). Note that as m runs over a complete residue system modulo m, so does am, and so G(n,χ) = m χ(am)e (nam) = χ(a) =0 χ(m)e (nam) = χ(a)g(na,χ). Seeing that /d = gcd(n,) n, we write n = u/d. Also write a = dv +. Thus, na = (u/d)(dv+) = uv+u/d = uv +n n (mod ), which implies that G(na,χ) = G(n,χ). Thus, G(n,χ) = χ(a)g(n,χ), and since G(n,χ) 0, we see that χ(a) =. We have seen that this is the criterion for χ to be imprimitive, and to be induced by a character mod d. This completes the proof. What s interesting to us is the contrapositive of Proposition : If χ is a primitive character mod and if gcd(n,) >, then G(n,χ) = 0. We exploit this result in a couple of ways. Corollary. If χ is a primitive character mod and n is any integer, then G(n,χ) = χ(n)g(,χ). Proof. We have seen that this holds if gcd(n,) >, since then both sides are 0. So assume that gcd(n,) =. In this case, the relation to be proved is equivalent to showing that χ(n)g(n,χ) = G(,χ), and we shall show this latter relation holds regardless if χ is primitive. As above, since n is coprime to, as m runs through a complete residue system mod, so does nm, so that G(,χ) = χ(nm)e (nm) = χ(n) χ(m)e (nm) = χ(n)g(n,χ). 5
6 Corollary 2. If χ is a primitive character mod then G(,χ) =. Proof. We have G(,χ) 2 = G(,χ)G(,χ) = G(,χ) χ(m)e ( m) = since Corollary implies that χ(m)g(,χ) = G(m,χ). Thus, G(,χ) 2 = χ(j)e (jm)e ( m) = χ(j) G(m,χ)e ( m), e (m(j )). We recognize the inner sum as 0 except in the case j = when the inner sum is. Thus, G(,χ) 2 = χ() =, which was to be proved. 5 The Pólya Vinogradov inequality Suppose χ is a character mod and we wish to get a reasonable upper estimate for S(χ,n), where S(χ,n) = χ(m). First note some simple cases: If χ is principal, then the sum is exactly the number of integers in [0,n] that are coprime to, and a simple upper estimate is n. But what if χ is not principal? Since S(χ,) = 0 in this case, we see that S(χ,n) is periodic with period, so we may assume that n. We always have the trivial estimate that S(χ,n) n. The Pólya Vinogradov inequality gives a nontrivial estimate for S(χ,n) if χ is not prinicipal and n is not too small compared with. First assume that χ is primitive and 3. (There are no primitive characters mod 2, and the only character mod is prinicpal.) As a periodic function with period, we have χ(m) = c j e (jm), where c j = χ(l)e ( lj) = G( lj,χ). l=0 Thus, S(χ,n) = χ(m) = c j e (jm) = c j e (jm). 6
7 This is neat because the inner sum is now a geometric progression and the numbers c j, being Gauss sums, have a nown magnitude via Corollary 2. In particular, we have S(χ,n) e (jm). j gcd(j,)= We now figure out what this geometric progression sums to: e (jm) = e (j(n+)) e (j) = e (j(n+)/2) e (j/2) e(j(n+)/2) e ( j(n+)/2). e (j/2) e ( j/2) Since e (θ) = for θ real, and since sin(2πθ/) = (e (θ) e ( θ))/2, we have e (jm) = sin(2πj(n+)/(2)) sin(2πj/(2)) sin(πj/). Thus, m= S(χ,n) j gcd(j,)= sin(πj/) 2 j /2 sin(πj/), using the symmetry sin(π θ) = sin(θ). Note that sinθ 2θ/π for 0 < θ π/2 (just draw a line from (0,0) to (π/2,) and see that sinθ lies at or above this line). Thus, / sin(πj/) (π/2)/(πj) = /j for j /2, and so 2 S(χ,n) j /2 j = We recognize this last sum as log(/2)+γ+o() as and to mae this numerical (that is, not involving a limit at infinity), note that if y 5, then j y j = Applying this for 0, we have j /2 6 j y j j /2 y 5 j. j < 37 log5+log(/2) < log, 60 dt t = log5+logy. since 37/60 < log0. Hence for 0 and χ a primitive character mod, we have S(χ,n)) < log (2) 7
8 for every integer n. One can chec directly for the 6 primitive characters with modulus in [3,9] and see that this result continues to be true there too. This then is the Pólya Vinogradov inequality. The result can be rather easily extended to non-principal characters that are not primitive, with a small sacrifice in the constant factor. (The constant factor is invisible in (2) since it is.) With considerably more wor, the Pólya Vinogradov inequality can be improved a little. When n is not too small, but smaller than, there is another inequality for S(χ,n), due to Burgess, that does better. But perhaps it is time to leave this where it is. 8
Arithmetic Statistics Lecture 1
Arithmetic Statistics Lecture 1 Álvaro Lozano-Robledo Department of Mathematics University of Connecticut May 28 th CTNT 2018 Connecticut Summer School in Number Theory Question What is Arithmetic Statistics?
More informationResearch Statement. Enrique Treviño. M<n N+M
Research Statement Enrique Treviño My research interests lie in elementary analytic number theory. Most of my work concerns finding explicit estimates for character sums. While these estimates are interesting
More informationThe Prime Number Theorem
Chapter 3 The Prime Number Theorem This chapter gives without proof the two basic results of analytic number theory. 3.1 The Theorem Recall that if f(x), g(x) are two real-valued functions, we write to
More informationBalanced subgroups of the multiplicative group
Balanced subgroups of the multiplicative group Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA Based on joint work with D. Ulmer To motivate the topic, let s begin with elliptic curves. If
More informationA numerically explicit Burgess inequality and an application to qua
A numerically explicit Burgess inequality and an application to quadratic non-residues Swarthmore College AMS Sectional Meeting Akron, OH October 21, 2012 Squares Consider the sequence Can it contain any
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 informationThe Least Inert Prime in a Real Quadratic Field
Explicit Palmetto Number Theory Series December 4, 2010 Explicit An upperbound on the least inert prime in a real quadratic field An integer D is a fundamental discriminant if and only if either D is squarefree,
More informationChapter 5. Number Theory. 5.1 Base b representations
Chapter 5 Number Theory The material in this chapter offers a small glimpse of why a lot of facts that you ve probably nown and used for a long time are true. It also offers some exposure to generalization,
More informationContest 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,
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 informationExplicit Bounds for the Burgess Inequality for Character Sums
Explicit Bounds for the Burgess Inequality for Character Sums INTEGERS, October 16, 2009 Dirichlet Characters Definition (Character) A character χ is a homomorphism from a finite abelian group G to C.
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 informationSmall gaps between primes
CRM, Université de Montréal Princeton/IAS Number Theory Seminar March 2014 Introduction Question What is lim inf n (p n+1 p n )? In particular, is it finite? Introduction Question What is lim inf n (p
More information(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
More informationALGEBRA. 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers
ALGEBRA CHRISTIAN REMLING 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers by Z = {..., 2, 1, 0, 1,...}. Given a, b Z, we write a b if b = ac for some
More informationFixed points for discrete logarithms
Fixed points for discrete logarithms Mariana Levin, U. C. Berkeley Carl Pomerance, Dartmouth College Suppose that G is a group and g G has finite order m. Then for each t g the integers n with g n = t
More informationPart II. Number Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler
More informationPRIME 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
More informationDefinition 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
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 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 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 informationBURGESS BOUND FOR CHARACTER SUMS. 1. Introduction Here we give a survey on the bounds for character sums of D. A. Burgess [1].
BURGESS BOUND FOR CHARACTER SUMS LIANGYI ZHAO 1. Introduction Here we give a survey on the bounds for character sums of D. A. Burgess [1]. We henceforth set (1.1) S χ (N) = χ(n), M
More informationMULTIPLICATIVE SEMIGROUPS RELATED TO THE 3x +1 PROBLEM. 1. Introduction The 3x +1iteration is given by the function on the integers for x even 3x+1
MULTIPLICATIVE SEMIGROUPS RELATED TO THE 3x + PROBLEM ANA CARAIANI Abstract. Recently Lagarias introduced the Wild semigroup, which is intimately connected to the 3x + Conjecture. Applegate and Lagarias
More informationAdvanced Number Theory Note #8: Dirichlet's theorem on primes in arithmetic progressions 29 August 2012 at 19:01
Advanced Number Theory Note #8: Dirichlet's theorem on primes in arithmetic progressions 29 August 2012 at 19:01 Public In this note, which is intended mainly as a technical memo for myself, I give a 'blow-by-blow'
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 informationDirichlet s Theorem. Martin Orr. August 21, The aim of this article is to prove Dirichlet s theorem on primes in arithmetic progressions:
Dirichlet s Theorem Martin Orr August 1, 009 1 Introduction The aim of this article is to prove Dirichlet s theorem on primes in arithmetic progressions: Theorem 1.1. If m, a N are coprime, then there
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 information3 The language of proof
3 The language of proof After working through this section, you should be able to: (a) understand what is asserted by various types of mathematical statements, in particular implications and equivalences;
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 informationTheorem 1.1 (Prime Number Theorem, Hadamard, de la Vallée Poussin, 1896). let π(x) denote the number of primes x. Then x as x. log x.
Chapter 1 Introduction 1.1 The Prime Number Theorem In this course, we focus on the theory of prime numbers. We use the following notation: we write f( g( as if lim f(/g( = 1, and denote by log the natural
More informationWORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...}
WORKSHEET ON NUMBERS, MATH 215 FALL 18(WHYTE) We start our study of numbers with the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } and their subset of natural numbers: N = {1, 2, 3,...} For now we will not
More informationWORKSHEET MATH 215, FALL 15, WHYTE. We begin our course with the natural numbers:
WORKSHEET MATH 215, FALL 15, WHYTE We begin our course with the natural numbers: N = {1, 2, 3,...} which are a subset of the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } We will assume familiarity with their
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
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 lim
More informationMath Real Analysis
1 / 28 Math 370 - Real Analysis G.Pugh Sep 3 2013 Real Analysis 2 / 28 3 / 28 What is Real Analysis? Wikipedia: Real analysis... has its beginnings in the rigorous formulation of calculus. It is a branch
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 informationMath 259: Introduction to Analytic Number Theory Primes in arithmetic progressions: Dirichlet characters and L-functions
Math 259: Introduction to Analytic Number Theory Primes in arithmetic progressions: Dirichlet characters and L-functions Dirichlet extended Euler s analysis from π(x) to π(x, a mod q) := #{p x : p is a
More informationComplex numbers, the exponential function, and factorization over C
Complex numbers, the exponential function, and factorization over C 1 Complex Numbers Recall that for every non-zero real number x, its square x 2 = x x is always positive. Consequently, R does not contain
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 informationPrimes in arithmetic progressions
(September 26, 205) Primes in arithmetic progressions Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/mfms/notes 205-6/06 Dirichlet.pdf].
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 informationA Few Primality Testing Algorithms
A Few Primality Testing Algorithms Donald Brower April 2, 2006 0.1 Introduction These notes will cover a few primality testing algorithms. There are many such, some prove that a number is prime, others
More informationReview Sheet for the Final Exam of MATH Fall 2009
Review Sheet for the Final Exam of MATH 1600 - Fall 2009 All of Chapter 1. 1. Sets and Proofs Elements and subsets of a set. The notion of implication and the way you can use it to build a proof. Logical
More informationFERMAT S TEST KEITH CONRAD
FERMAT S TEST KEITH CONRAD 1. Introduction Fermat s little theorem says for prime p that a p 1 1 mod p for all a 0 mod p. A naive extension of this to a composite modulus n 2 would be: for all a 0 mod
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 informationProblem 1. Let I and J be ideals in a ring commutative ring R with 1 R. Recall
I. Take-Home Portion: Math 350 Final Exam Due by 5:00pm on Tues. 5/12/15 No resources/devices other than our class textbook and class notes/handouts may be used. You must work alone. Choose any 5 problems
More informationFourier Analysis. 1 Fourier basics. 1.1 Examples. 1.2 Characters form an orthonormal basis
Fourier Analysis Topics in Finite Fields (Fall 203) Rutgers University Swastik Kopparty Last modified: Friday 8 th October, 203 Fourier basics Let G be a finite abelian group. A character of G is simply
More informationDiscrete Logarithms. Let s begin by recalling the definitions and a theorem. Let m be a given modulus. Then the finite set
Discrete Logarithms Let s begin by recalling the definitions and a theorem. Let m be a given modulus. Then the finite set Z/mZ = {[0], [1],..., [m 1]} = {0, 1,..., m 1} of residue classes modulo m is called
More informationA Combinatorial Approach to Finding Dirichlet Generating Function Identities
The Waterloo Mathematics Review 3 A Combinatorial Approach to Finding Dirichlet Generating Function Identities Alesandar Vlasev Simon Fraser University azv@sfu.ca Abstract: This paper explores an integer
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 informationPATTERNS OF PRIMES IN ARITHMETIC PROGRESSIONS
PATTERNS OF PRIMES IN ARITHMETIC PROGRESSIONS JÁNOS PINTZ Rényi Institute of the Hungarian Academy of Sciences CIRM, Dec. 13, 2016 2 1. Patterns of primes Notation: p n the n th prime, P = {p i } i=1,
More informationPrime Gaps and Adeles. Tom Wright Johns Hopkins University (joint with B. Weiss, University of Michigan)
Prime Gaps and Adeles Tom Wright Johns Hopkins University (joint with B. Weiss, University of Michigan) January 16, 2010 History 2000 years ago, Euclid posed the following conjecture: Twin Prime Conjecture:
More informationCHAPTER 3. Congruences. Congruence: definitions and properties
CHAPTER 3 Congruences Part V of PJE Congruence: definitions and properties Definition. (PJE definition 19.1.1) Let m > 0 be an integer. Integers a and b are congruent modulo m if m divides a b. We write
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 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 informationWilson s theorem for block designs Talks given at LSBU June August 2014 Tony Forbes
Wilson s theorem for block designs Talks given at LSBU June August 2014 Tony Forbes Steiner systems S(2, k, v) For k 3, a Steiner system S(2, k, v) is usually defined as a pair (V, B), where V is a set
More information8 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.
More informationJournal of Combinatorics and Number Theory 1(2009), no. 1, ON SUMS OF PRIMES AND TRIANGULAR NUMBERS. Zhi-Wei Sun
Journal of Combinatorics and Number Theory 1(009), no. 1, 65 76. ON SUMS OF PRIMES AND TRIANGULAR NUMBERS Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing 10093, People s Republic of China
More informationCongruent Number Problem and Elliptic curves
Congruent Number Problem and Elliptic curves December 12, 2010 Contents 1 Congruent Number problem 2 1.1 1 is not a congruent number.................................. 2 2 Certain Elliptic Curves 4 3 Using
More information7. 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.
More informationax b mod m. has a solution if and only if d b. In this case, there is one solution, call it x 0, to the equation and there are d solutions x m d
10. Linear congruences In general we are going to be interested in the problem of solving polynomial equations modulo an integer m. Following Gauss, we can work in the ring Z m and find all solutions to
More informationPossible Group Structures of Elliptic Curves over Finite Fields
Possible Group Structures of Elliptic Curves over Finite Fields Igor Shparlinski (Sydney) Joint work with: Bill Banks (Columbia-Missouri) Francesco Pappalardi (Roma) Reza Rezaeian Farashahi (Sydney) 1
More informationIrredundant Families of Subcubes
Irredundant Families of Subcubes David Ellis January 2010 Abstract We consider the problem of finding the maximum possible size of a family of -dimensional subcubes of the n-cube {0, 1} n, none of which
More informationNONABELIAN GROUPS WITH PERFECT ORDER SUBSETS
NONABELIAN GROUPS WITH PERFECT ORDER SUBSETS CARRIE E. FINCH AND LENNY JONES Abstract. Let G be a finite group and let x G. Define the order subset of G determined by x to be the set of all elements in
More information1. 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.
More informationOn the second smallest prime non-residue
On the second smallest prime non-residue Kevin J. McGown 1 Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093 Abstract Let χ be a non-principal Dirichlet
More informationBefore giving the detailed proof, we outline our strategy. Define the functions. for Re s > 1.
Chapter 7 The Prime number theorem for arithmetic progressions 7. The Prime number theorem We denote by π( the number of primes. We prove the Prime Number Theorem. Theorem 7.. We have π( as. log Before
More informationMath 259: Introduction to Analytic Number Theory Exponential sums IV: The Davenport-Erdős and Burgess bounds on short character sums
Math 259: Introduction to Analytic Number Theory Exponential sums IV: The Davenport-Erdős and Burgess bounds on short character sums Finally, we consider upper bounds on exponential sum involving a character,
More informationC.T.Chong National University of Singapore
NUMBER THEORY AND THE DESIGN OF FAST COMPUTER ALGORITHMS C.T.Chong National University of Singapore The theory of numbers has long been considered to be among the purest of pure mathematics. Gauss ( 1777-1855)
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 informationNotes: Pythagorean Triples
Math 5330 Spring 2018 Notes: Pythagorean Triples Many people know that 3 2 + 4 2 = 5 2. Less commonly known are 5 2 + 12 2 = 13 2 and 7 2 + 24 2 = 25 2. Such a set of integers is called a Pythagorean Triple.
More informationCHAPTER 1. REVIEW: NUMBERS
CHAPTER. REVIEW: NUMBERS Yes, mathematics deals with numbers. But doing math is not number crunching! Rather, it is a very complicated psychological process of learning and inventing. Just like listing
More informationFinal Exam Review. 2. Let A = {, { }}. What is the cardinality of A? Is
1. Describe the elements of the set (Z Q) R N. Is this set countable or uncountable? Solution: The set is equal to {(x, y) x Z, y N} = Z N. Since the Cartesian product of two denumerable sets is denumerable,
More informationElliptic Curves Spring 2013 Lecture #12 03/19/2013
18.783 Elliptic Curves Spring 2013 Lecture #12 03/19/2013 We now consider our first practical application of elliptic curves: factoring integers. Before presenting the elliptic curve method (ECM) for factoring
More informationarxiv: v1 [math.nt] 31 Dec 2018
POSITIVE PROPORTION OF SORT INTERVALS CONTAINING A PRESCRIBED NUMBER OF PRIMES DANIELE MASTROSTEFANO arxiv:1812.11784v1 [math.nt] 31 Dec 2018 Abstract. We will prove that for every m 0 there exists an
More informationThe group (Z/nZ) February 17, In these notes we figure out the structure of the unit group (Z/nZ) where n > 1 is an integer.
The group (Z/nZ) February 17, 2016 1 Introduction In these notes we figure out the structure of the unit group (Z/nZ) where n > 1 is an integer. If we factor n = p e 1 1 pe, where the p i s are distinct
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More informationChapter 4. Characters and Gauss sums. 4.1 Characters on finite abelian groups
Chapter 4 Characters and Gauss sums 4.1 Characters on finite abelian groups In what follows, abelian groups are multiplicatively written, and the unit element of an abelian group A is denoted by 1 or 1
More informationNOTES ON ZHANG S PRIME GAPS PAPER
NOTES ON ZHANG S PRIME GAPS PAPER TERENCE TAO. Zhang s results For any natural number H, let P (H) denote the assertion that there are infinitely many pairs of distinct primes p, q with p q H; thus for
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 informationM381 Number Theory 2004 Page 1
M81 Number Theory 2004 Page 1 [[ Comments are written like this. Please send me (dave@wildd.freeserve.co.uk) details of any errors you find or suggestions for improvements. ]] Question 1 20 = 2 * 10 +
More informationp-adic L-functions for Dirichlet characters
p-adic L-functions for Dirichlet characters Rebecca Bellovin 1 Notation and conventions Before we begin, we fix a bit of notation. We mae the following convention: for a fixed prime p, we set q = p if
More informationLecture 11: Cantor-Zassenhaus Algorithm
CS681 Computational Number Theory Lecture 11: Cantor-Zassenhaus Algorithm Instructor: Piyush P Kurur Scribe: Ramprasad Saptharishi Overview In this class, we shall look at the Cantor-Zassenhaus randomized
More information1, for s = σ + it where σ, t R and σ > 1
DIRICHLET L-FUNCTIONS AND DEDEKIND ζ-functions FRIMPONG A. BAIDOO Abstract. We begin by introducing Dirichlet L-functions which we use to prove Dirichlet s theorem on arithmetic progressions. From there,
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 informationClassification of Finite Fields
Classification of Finite Fields In these notes we use the properties of the polynomial x pd x to classify finite fields. The importance of this polynomial is explained by the following basic proposition.
More informationChapter 7. Number Theory. 7.1 Prime Numbers
Chapter 7 Number Theory 7.1 Prime Numbers Any two integers can be multiplied together to produce a new integer. For example, we can multiply the numbers four and five together to produce twenty. In this
More informationNumber Theory. Zachary Friggstad. Programming Club Meeting
Number Theory Zachary Friggstad Programming Club Meeting Outline Factoring Sieve Multiplicative Functions Greatest Common Divisors Applications Chinese Remainder Theorem Throughout, problems to try are
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 information2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.
2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say
More informationMATH Fundamental Concepts of Algebra
MATH 4001 Fundamental Concepts of Algebra Instructor: Darci L. Kracht Kent State University April, 015 0 Introduction We will begin our study of mathematics this semester with the familiar notion of even
More informationPrime Numbers and Irrational Numbers
Chapter 4 Prime Numbers and Irrational Numbers Abstract The question of the existence of prime numbers in intervals is treated using the approximation of cardinal of the primes π(x) given by Lagrange.
More informationIB Mathematics HL Year 2 Unit 11: Completion of Algebra (Core Topic 1)
IB Mathematics HL Year Unit : Completion of Algebra (Core Topic ) Homewor for Unit Ex C:, 3, 4, 7; Ex D: 5, 8, 4; Ex E.: 4, 5, 9, 0, Ex E.3: (a), (b), 3, 7. Now consider these: Lesson 73 Sequences and
More informationSUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by
SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples
More informationKUMMER S LEMMA KEITH CONRAD
KUMMER S LEMMA KEITH CONRAD Let p be an odd prime and ζ ζ p be a primitive pth root of unity In the ring Z[ζ], the pth power of every element is congruent to a rational integer mod p, since (c 0 + c 1
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 informationNew bounds on gaps between primes
New bounds on gaps between primes Andrew V. Sutherland Massachusetts Institute of Technology Brandeis-Harvard-MIT-Northeastern Joint Colloquium October 17, 2013 joint work with Wouter Castryck, Etienne
More informationSmall gaps between prime numbers
Small gaps between prime numbers Yitang Zhang University of California @ Santa Barbara Mathematics Department University of California, Santa Barbara April 20, 2016 Yitang Zhang (UCSB) Small gaps between
More information