DOCSLIB.ORG

Using Dynamical Systems to Construct Infinitely Many Primes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Using Dynamical Systems to Construct Infinitely Many Primes Mathematical Assoc. of America American Mathematical Monthly 121:1 August 24, 2017 12:20 a.m. MonthlyArticleDynamicsOnly.v5.tex page 1 Using Dynamical Systems to Construct Infinitely Many Primes Andrew Granville Abstract. Euclid’s proof can be reworked to construct infinitely many primes, in many differ- ent ways, using ideas from arithmetic dynamics. 1. CONSTRUCTING INFINITELY MANY PRIMES. The first known proof that there are infinitely many primes, which appears in Euclid’s Elements, is a proof by contradiction. This proof does not indicate how to find primes, despite establishing that there are infinitely many of them. However a minor variant does do so. The key to either proof is the theorem that Every integer q > 1 has a prime factor. (This was first formally proved by Euclid on the way to establishing the fundamental theorem of arithmetic.) The idea is to Construct an infinite sequence of distinct, pairwise coprime, integers a0, a1,..., that is, a sequence for which gcd(am, an) = 1 whenever m 6= n. We then obtain infinitely many primes, the prime factors of the an, as proved in the following result. Proposition 1. Suppose that a0, a1,... is an infinite sequence of distinct, pairwise coprime, integers. Let pn be a prime divisor of an whenever |an| > 1. Then the pn form an infinite sequence of distinct primes. Proof. The pn are distinct for if not, then pm = pn for some m 6= n and so pm = gcd(pm,pn) divides gcd(am, an) = 1, a contradiction. As the an are distinct, any given value (in particular, −1, 0 and 1) can be attained at most once, and therefore all but at most three of the an have absolute value > 1, and so have a prime factor pn. To construct such a sequence by modifying Euclid’s proof, let E0 = 2 and arXiv:1708.06953v1 [math.NT] 23 Aug 2017 En = E0E1 · · · En−1 + 1 for each n ≥ 1. If m<n, then Em divides E0E1 · · · En−1 = En − 1 (as Em is one of the terms in the product) and so gcd(Em, En) divides gcd(En − 1, En) = 1, which implies that gcd(Em, En) = 1. Therefore, if pn is a prime divisor of En for each n ≥ 0, then p0,p1,... is an infinite sequence of distinct primes. 2n The Fermat numbers, F0, F1,..., defined by Fn = 2 + 1 for each n ≥ 0, are a more familiar sequence of pairwise coprime integers. Fermat had actually conjectured that the Fn are all primes. His claim starts off correct: 3, 5, 17, 257, 65537 are all prime, but is false for F5 = 641 × 6700417, as Euler famously noted. It is an open January 2014] DYNAMICAL SYSTEMS AND PRIMES 1 Mathematical Assoc. of America American Mathematical Monthly 121:1 August 24, 2017 12:20 a.m. MonthlyArticleDynamicsOnly.v5.tex page 2 1 question as to whether there are infinitely many primes of the form Fn. Nonetheless we can prove that the Fn are pairwise coprime in a similar way to the En, using the identity Fn = F0F1 · · · Fn−1 + 2 for each n ≥ 1 and the fact that the Fn are all odd. We deduce from Proposition 1 that if pn is a prime divisor of Fn for each n ≥ 0, then p0,p1,... is an infinite sequence of distinct primes.2 For any sequence of integers (an)n≥0 we call pn a private prime factor of an if it divides an and no other am. The primes pn constructed in Proposition 1 are therefore each private prime factors of the an. 2. A SIMPLER FORMULATION. In Section 1, the sequences (En)n≥0 and (Fn)n≥0 were constructed in a similar way, multiplying all of the terms of the se- quence so far together and adding a constant. We would like a simpler, unified way to view these two sequences, with an eye to generalization. This is not difficult because the recurrence for the En can be rewritten as En+1 = E0E1 · · · En−1 · En +1=(En − 1)En +1= f(En), 2 where f(x)= x − x + 1. Similarly the Fn-values can be determined by 2n 2n Fn+1 = (2 + 1)(2 − 1)+2= Fn(Fn − 2)+1= f(Fn), 2 where f(x) = x − 2x + 2. So they are both examples of sequences (xn)n≥0 for which xn+1 = f(xn) for some polynomial f(x) ∈ Z[x]. The terms of the sequence are all given by the recursive formula, so that n xn = f(f(...f(x0))) = f (x0), n times | {z } where the notation f n denotes the polynomial obtained by composing f with itself n times (which is definitely not the nth power of f). Any such sequence (xn)n≥0 is called the orbit of x0 under the map f, since the sequence is completely determined once one knows x0 and f. We sometimes write the orbit as x0 → x1 → x2 →··· . The key to the proof that the En’s are pairwise coprime is that En ≡ 1 (mod Em) whenever n>m ≥ 0; and the key to the proof that the Fn’s are pairwise coprime is that Fn ≡ 2 (mod Fm) whenever n>m ≥ 0. These congruences are not difficult to deduce using the following lemma from elementary number theory. 1 The only Fermat numbers known to be primes have n ≤ 4. We know that the Fn are composite for 5 ≤ n ≤ 30 and for many other n besides. It is always a significant moment when a Fermat number is factored for the first time. It could be that all Fn with n > 4 are composite, or they might all be prime from some sufficiently large n onwards, or some might be prime and some composite. Currently, we have no way of knowing which is true. 2Goldbach was the first to use the prime divisors of the Fermat numbers to prove that there are infinitely many primes, in a letter to Euler in late July 1730. 2 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121 Mathematical Assoc. of America American Mathematical Monthly 121:1 August 24, 2017 12:20 a.m. MonthlyArticleDynamicsOnly.v5.tex page 3 Lemma 1. Let g(x) be a polynomial with integer coefficients. For any integers a 6= b, their difference a − b divides g(a) − g(b). Moreover, if m is an integer for which a ≡ b (mod m), then g(a) ≡ g(b) (mod m). For f(x)= x2 − x + 1, the orbit of 0 under the map f is 0 → 1 → 1 → ··· . Therefore if n>m ≥ 0, then we have n−m n−m En = f (Em) ≡ f (0) = 1 (mod Em), and so (En, Em) = (1, Em) = 1. (For clarity, we have used an “=” sign when two numbers are equal despite working with a congruence. Typically we use the notation (a, b) rather than gcd(a, b).) Similarly if f(x) = x2 − 2x + 2, then the orbit of 0 n−m n−m under the map f is 0 → 2 → 2 → ··· and so Fn = f (Fm) ≡ f (0) = 2 (mod Fm), which implies that if n>m, then (Fn, Fm) = (2, Fm) = 1. This reformulation of two of the best-known proofs of the infinitude of primes hints at the possibility of a more general approach. 3. DIFFERENT STARTING POINTS. We just saw that the orbit of 2 under the map x → x2 − x + 1 is an infinite sequence of pairwise coprime integers, 2 → 3 → 7 → 43 → 1807 →··· . What about other orbits? The orbit of 4 is also an infinite sequence of pairwise coprime integers beginning 4 → 13 → 157 → 24493 →··· , as is the orbit of 5 which begins 5 → 21 → 421 →··· . The same proof as before yields that no two integers in a given orbit have a common factor. The orbit of 3 under the the map x → x2 − 2x + 2 yielded the Fermat numbers 3 → 5 → 17 → 257 →··· , but starting at 4 we get 4 → 10 → 82 → 6562 →··· . These are obviously not pairwise prime as every number in the orbit is even, but if we divide through by 2, then we get 2 → 5 → 41 → 3281 →··· 2 which are pairwise coprime. To prove this, note that x0 = 4 and xn+1 = xn − 2xn + 2 for all n ≥ 0, and so the above proof yields that if m<n then (xm,xn)= (xm, 2) = 2. Therefore, taking an = xn/2 for every n we deduce that (am, an) = (xm/2,xn/2) = (xm/2, 1) = 1. This same idea works for every orbit under this map: we get an infinite sequence of pairwise coprime integers by dividing through by 1 or 2, depending on whether x0 is odd or even. However, things can be more complicated: Consider the orbit of x0 = 3 under the map x → x2 − 6x − 1. We have 3 →−10 → 159 → 24326 → 591608319 →··· . Here xn is divisible by 3 if n is even, and is divisible by 2 if n is odd. If we let an = xn/3 when n is even, and an = xn/2 when n is odd, then one can show the terms of the resulting sequence, 1 →−5 → 53 → 12163 → 197202773 →··· , are indeed pairwise coprime. Another surprising example is given by the orbit of 6 under the map x → 7+ x5(x − 1)(x − 7). Reducing the elements of the orbit mod 7 we find that 6 → 5 → 4 → 3 → 2 → 1 → 0 → 0 →··· (mod 7), January 2014] DYNAMICAL SYSTEMS AND PRIMES 3 Mathematical Assoc. of America American Mathematical Monthly 121:1 August 24, 2017 12:20 a.m. MonthlyArticleDynamicsOnly.v5.tex page 4 as x5(x − 1)(x − 7) ≡ x6(x − 1) (mod 7), which is ≡ x − 1 (mod 7) if x 6≡ 0 (mod 7) by Fermat’s little theorem.
Load more

Recommended publications
  • On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function
    mathematics Article On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function ŠtˇepánHubálovský 1,* and Eva Trojovská 2 1 Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic 2 Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +420-49-333-2704 Received: 3 October 2020; Accepted: 14 October 2020; Published: 16 October 2020 Abstract: Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk ≡ 0 (mod n). In this paper, we shall find all positive solutions of the Diophantine equation z(j(n)) = n, where j is the Euler totient function. Keywords: Fibonacci numbers; order of appearance; Euler totient function; fixed points; Diophantine equations MSC: 11B39; 11DXX 1. Introduction Let (Fn)n≥0 be the sequence of Fibonacci numbers which is defined by 2nd order recurrence Fn+2 = Fn+1 + Fn, with initial conditions Fi = i, for i 2 f0, 1g. These numbers (together with the sequence of prime numbers) form a very important sequence in mathematics (mainly because its unexpectedly and often appearance in many branches of mathematics as well as in another disciplines). We refer the reader to [1–3] and their very extensive bibliography. We recall that an arithmetic function is any function f : Z>0 ! C (i.e., a complex-valued function which is defined for all positive integer).
    [Show full text]
  • Lecture 3: Chinese Remainder Theorem 25/07/08 to Further
    Department of Mathematics MATHS 714 Number Theory: Lecture 3: Chinese Remainder Theorem 25/07/08 To further reduce the amount of computations in solving congruences it is important to realise that if m = m1m2 ...ms where the mi are pairwise coprime, then a ≡ b (mod m) if and only if a ≡ b (mod mi) for every i, 1 ≤ i ≤ s. This leads to the question of simultaneous solution of a system of congruences. Theorem 1 (Chinese Remainder Theorem). Let m1,m2,...,ms be pairwise coprime integers ≥ 2, and b1, b2,...,bs arbitrary integers. Then, the s congruences x ≡ bi (mod mi) have a simultaneous solution that is unique modulo m = m1m2 ...ms. Proof. Let ni = m/mi; note that (mi, ni) = 1. Every ni has an inversen ¯i mod mi (lecture 2). We show that x0 = P1≤j≤s nj n¯j bj is a solution of our system of s congruences. Since mi divides each nj except for ni, we have x0 = P1≤j≤s njn¯j bj ≡ nin¯ibj (mod mi) ≡ bi (mod mi). Uniqueness: If x is any solution of the system, then x − x0 ≡ 0 (mod mi) for all i. This implies that m|(x − x0) i.e. x ≡ x0 (mod m). Exercise 1. Find all solutions of the system 4x ≡ 2 (mod 6), 3x ≡ 5 (mod 7), 2x ≡ 4 (mod 11). Exercise 2. Using the Chinese Remainder Theorem (CRT), solve 3x ≡ 11 (mod 2275). Systems of linear congruences in one variable can often be solved efficiently by combining inspection (I) and Euclid’s algorithm (EA) with the Chinese Remainder Theorem (CRT).
    [Show full text]
  • Sequences of Primes Obtained by the Method of Concatenation
    SEQUENCES OF PRIMES OBTAINED BY THE METHOD OF CONCATENATION (COLLECTED PAPERS) Copyright 2016 by Marius Coman Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721 Peer-Reviewers: Dr. A. A. Salama, Faculty of Science, Port Said University, Egypt. Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Mohamed Eisa, Dept. of Computer Science, Port Said Univ., Egypt. EAN: 9781599734668 ISBN: 978-1-59973-466-8 1 INTRODUCTION The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two numbers by their numerals. That is, the concatenation of 69 and 420 is 69420”. Though the method of concatenation is widely considered as a part of so called “recreational mathematics”, in fact this method can often lead to very “serious” results, and even more than that, to really amazing results. This is the purpose of this book: to show that this method, unfairly neglected, can be a powerful tool in number theory. In particular, as revealed by the title, I used the method of concatenation in this book to obtain possible infinite sequences of primes. Part One of this book, “Primes in Smarandache concatenated sequences and Smarandache-Coman sequences”, contains 12 papers on various sequences of primes that are distinguished among the terms of the well known Smarandache concatenated sequences (as, for instance, the prime terms in Smarandache concatenated odd
    [Show full text]
  • Number Theory Course Notes for MA 341, Spring 2018
    Number Theory Course notes for MA 341, Spring 2018 Jared Weinstein May 2, 2018 Contents 1 Basic properties of the integers 3 1.1 Definitions: Z and Q .......................3 1.2 The well-ordering principle . .5 1.3 The division algorithm . .5 1.4 Running times . .6 1.5 The Euclidean algorithm . .8 1.6 The extended Euclidean algorithm . 10 1.7 Exercises due February 2. 11 2 The unique factorization theorem 12 2.1 Factorization into primes . 12 2.2 The proof that prime factorization is unique . 13 2.3 Valuations . 13 2.4 The rational root theorem . 15 2.5 Pythagorean triples . 16 2.6 Exercises due February 9 . 17 3 Congruences 17 3.1 Definition and basic properties . 17 3.2 Solving Linear Congruences . 18 3.3 The Chinese Remainder Theorem . 19 3.4 Modular Exponentiation . 20 3.5 Exercises due February 16 . 21 1 4 Units modulo m: Fermat's theorem and Euler's theorem 22 4.1 Units . 22 4.2 Powers modulo m ......................... 23 4.3 Fermat's theorem . 24 4.4 The φ function . 25 4.5 Euler's theorem . 26 4.6 Exercises due February 23 . 27 5 Orders and primitive elements 27 5.1 Basic properties of the function ordm .............. 27 5.2 Primitive roots . 28 5.3 The discrete logarithm . 30 5.4 Existence of primitive roots for a prime modulus . 30 5.5 Exercises due March 2 . 32 6 Some cryptographic applications 33 6.1 The basic problem of cryptography . 33 6.2 Ciphers, keys, and one-time pads .
    [Show full text]
  • Introduction to Number Theory CS1800 Discrete Math; Notes by Virgil Pavlu; Updated November 5, 2018
    Introduction to Number Theory CS1800 Discrete Math; notes by Virgil Pavlu; updated November 5, 2018 1 modulo arithmetic All numbers here are integers. The integer division of a at n > 1 means finding the unique quotient q and remainder r 2 Zn such that a = nq + r where Zn is the set of all possible remainders at n : Zn = f0; 1; 2; 3; :::; n−1g. \mod n" = remainder at division with n for n > 1 (n it has to be at least 2) \a mod n = r" means mathematically all of the following : · r is the remainder of integer division a to n · a = n ∗ q + r for some integer q · a; r have same remainder when divided by n · a − r = nq is a multiple of n · n j a − r, a.k.a n divides a − r EXAMPLES 21 mod 5 = 1, because 21 = 5*4 +1 same as saying 5 j (21 − 1) 24 = 10 = 3 = -39 mod 7 , because 24 = 7*3 +3; 10=7*1+3; 3=7*0 +3; -39=7*(-6)+3. Same as saying 7 j (24 − 10) or 7 j (3 − 10) or 7 j (10 − (−39)) etc LEMMA two numbers a; b have the same remainder mod n if and only if n divides their difference. We can write this in several equivalent ways: · a mod n = b mod n, saying a; b have the same remainder (or modulo) · a = b( mod n) · n j a − b saying n divides a − b · a − b = nk saying a − b is a multiple of n (k is integer but its value doesnt matter) 1 EXAMPLES 21 = 11 (mod 5) = 1 , 5 j (21 − 11) , 21 mod 5 = 11 mod 5 86 mod 10 = 1126 mod 10 , 10 j (86 − 1126) , 86 − 1126 = 10k proof: EXERCISE.
    [Show full text]
  • On Sets of Coprime Integers in Intervals Paul Erdös, András Sárközy
    On sets of coprime integers in intervals Paul Erdös, András Sárközy To cite this version: Paul Erdös, András Sárközy. On sets of coprime integers in intervals. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1993, 16, pp.1 - 20. hal-01108688 HAL Id: hal-01108688 https://hal.archives-ouvertes.fr/hal-01108688 Submitted on 23 Jan 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1 Hardy-Ramanujan Joumal Vol.l6 (1993) 1-20 On sets of coprime integers in intervals P. Erdos and Sarkozy 1 1. Throughout this paper we use the following notations : ~ denotes the set of the integers. N denotes the set of the positive integers. For A C N,m E N,u E ~we write .A(m,u) ={f.£; a E A, a= u(mod m)}. IP(n) denotes Euler';; function. Pk dznotcs the kth prime: P1 = 2,p2 = 3,· ··and k we put Pk = IJP•· If k E N and k ~ 2, then .PA:(A) denotes the number of i=l the k-tuples (at,···, ak) such that at E A,·· · , ak E A,ai < a2 < · · · < ak and (at,aj = 1) for 1::; i < j::; k.
    [Show full text]
  • Dynamical Systems and Infinite Sequences of Coprime Integers
    Dynamical systems and infinite sequences of coprime integers Max van Spengler 3935892 Supervisor: Damaris Schindler Departement Wiskunde Universiteit Utrecht Nederland 31-05-2018 Contents 1 Introduction 2 1.1 Goldbach's proof . 3 1.2 Erd}os'proof . 4 1.3 Furstenberg's proof . 5 2 Theoretical background 7 2.1 Number theory . 7 2.2 Dynamical systems . 10 2.3 Additional theory . 10 3 Constructing infinitely many primes 16 3.1 A general method when 0 is strictly periodic . 17 3.2 General methods when 0 is periodic . 21 3.3 The case where 0 has a wandering orbit . 25 4 New results 29 4.1 A restriction on the weak abc-conjecture . 29 4.2 Applying the exponential bound . 30 5 Summary 32 1 1 Introduction Since the Hellenistic period, prime numbers and their properties have been a fundamental object of interest in mathematics [MB11]. The defining property of such a number is that it is only divisible by 1 and itself. Their counterparts are the composite numbers. These are numbers defined as being divisible by at least one number other than 1 or themselves. One of the earliest results obtained from research into these numbers is now known as the fundamental theorem of arithmetic. Fundamental theorem of arithmetic. Every natural number > 1 is either prime or can be written, up to the order of factors, as a unique product of primes. Proof. We will give a proof by induction. First, note that 2 and 3 are prime. Now assume the theorem to be true for all numbers between 1 and n + 1 for some n 2 N, so each number in between can be written as a unique product of primes.
    [Show full text]
  • Theory of Numbers
    Unit 1 Theory of Numbers Natural Numbers Thenumbers1,2,3,…,whichareusedincountingarecalled naturalnumbers or positiveintegers. Basic Properties of Natural Numbers In the system of natural numbers, we have two ‘operations’ addition and multiplication with the followingproperties. Let x,, y z denotearbitrarynaturalnumbers,then 1. x+ y isanaturalnumber i.e., thesumoftwonaturalnumberisagainanaturalnumber. 2. Commutative law of addition x+ y = y + x 3. Associative law of addition x+()() y + z = x + y + z 4. x⋅ y isanaturalnumber i.e., productoftwonaturalnumbersisanaturalnumber. 5. Commutative law of multiplication x⋅ y = y ⋅ x 6. Associative law of multiplication x⋅()() y ⋅ z = x ⋅ y ⋅ z 7. Existence of multiplicative identity a⋅1 = a 8. Distributive law x() y+ z = xy + xz Divisibility of Integers An integerx ≠ 0 divides y, if there exists an integer a such that y= ax and thus we write as x| y ( x divides y). If x doesnotdivides y,wewriteas x\| y (x doesnotdivides y ) [Thiscanalsobestatedas y isdivisibleby x or x isadivisorof y or y isamultipleof x]. Properties of Divisibility 1. x | y and y | z ⇒ x | z 2. x | y and x | z ⇒x| ( ky± lz ) forall k, l∈ z. z = setofallintegers. 3. x | y and y | x ⇒ x= ± y 4. x | y,where x>0, y > 0 ⇒x≤ y 5. x | y ⇒ x | yz foranyinteger z. 6. x | y iff nx | ny,where n ≠ 0. 2 IndianNational MathematicsOlympiad Test of Divisibility Divisibility by certain special numbers can be determined without actually carrying out the process of division.Thefollowingtheoremsummarizestheresult: ApositiveintegerNisdivisibleby • 2ifandonlyifthelastdigit(unit'sdigit)iseven.
    [Show full text]
  • Number Theory
    Chapter 3 Number Theory Part of G12ALN Contents Algebra and Number Theory G12ALN cw '17 0 Review of basic concepts and theorems The contents of this first section { well zeroth section, really { is mostly repetition of material from last year. Notations: N = f1; 2; 3;::: g, Z = f:::; −2; −1; 0; 1; 2;::: g. If A is a finite set, I write #A for the number of elements in A. Theorem 0.1 (Long division). If a; b 2 Z and b > 0, then there are unique integers q and r such that a = q b + r with 0 6 r < b: Proof. Theorem 3.2 in G11MSS. The integer q is called the quotient and r the remainder. We say that b divides a if the remainder is zero. It will be denoted by b j a. There is an interesting variant to this: There are unique integers q0 and 0 0 0 b 0 b 0 r with a = q b + r and − 2 < r 6 2 . Instead of remainder, r is called the least residue of a modulo b. Example. Take a = 62 and b = 9. Then the quotient is q = 6 and the remainder is r = 8. The least residue is r0 = −1. 0.1 The greatest common divisor §Definition. Let a and b be integers not both equal to 0. The greatest ¤ common divisor of a and b is the largest integer dividing both a and b. We will denote it by (a; b). For convenience, we set (0; 0) = 0. ¦ ¥ Let a; b 2 Z.
    [Show full text]
  • The Probability That Two Random Integers Are Coprime Julien Bureaux, Nathanaël Enriquez
    The probability that two random integers are coprime Julien Bureaux, Nathanaël Enriquez To cite this version: Julien Bureaux, Nathanaël Enriquez. The probability that two random integers are coprime. 2016. hal-01413829 HAL Id: hal-01413829 https://hal.archives-ouvertes.fr/hal-01413829 Preprint submitted on 11 Dec 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THE PROBABILITY THAT TWO RANDOM INTEGERS ARE COPRIME JULIEN BUREAUX AND NATHANAEL¨ ENRIQUEZ Abstract. An equivalence is proven between the Riemann Hypothesis and the speed of con- vergence to 6/π2 of the probability that two independent random variables following the same geometric distribution are coprime integers, when the parameter of the distribution goes to 0. What is the probability that two random integers are coprime? Since there is no uniform probability distribution on N, this question must be made more precise. Before going into probabilistic considerations, we would like to remind a seminal result of Dirichlet [1]: the set of ordered pairs of coprime integers has an asymptotic density in N2 which is equal to 6/π2 0.P608. The proof is not trivial and can be found in the book of Hardy and Wright ([2], Theorem≈ 332), but if one admits the existence of a density δ, its value can be easily obtained.
    [Show full text]
  • Integer Factoring
    Designs, Codes and Cryptography, 19, 101–128 (2000) c 2000 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Integer Factoring ARJEN K. LENSTRA [email protected] Citibank, N.A., 1 North Gate Road, Mendham, NJ 07945-3104, USA Abstract. Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization. Keywords: Integer factorization, quadratic sieve, number field sieve, elliptic curve method, Morrison–Brillhart Approach 1. Introduction Factoring a positive integer n means finding positive integers u and v such that the product of u and v equals n, and such that both u and v are greater than 1. Such u and v are called factors (or divisors)ofn, and n = u v is called a factorization of n. Positive integers that can be factored are called composites. Positive integers greater than 1 that cannot be factored are called primes. For example, n = 15 can be factored as the product of the primes u = 3 and v = 5, and n = 105 can be factored as the product of the prime u = 7 and the composite v = 15. A factorization of a composite number is not necessarily unique: n = 105 can also be factored as the product of the prime u = 5 and the composite v = 21. But the prime factorization of a number—writing it as a product of prime numbers—is unique, up to the order of the factors: n = 3 5 7isthe prime factorization of n = 105, and n = 5 is the prime factorization of n = 5.
    [Show full text]
  • Prime Factorization and Cryptography a Theoretical Introduction to the General Number Field Sieve
    Prime Factorization and Cryptography A theoretical introduction to the General Number Field Sieve Barry van Leeuwen University of Bristol 10 CP Undergraduate Project Supervisor: Dr. Tim Dokchitser February 1, 2019 Acknowledgement of Sources For all ideas taken from other sources (books, articles, internet), the source of the ideas is mentioned in the main text and fully referenced at the end of the report. All material which is quoted essentially word-for-word from other sources is given in quotation marks and referenced. Pictures and diagrams copied from the internet or other sources are labelled with a reference to the web page,book, article etc. Signed: Barry van Leeuwen Dated: February 1, 2019 Abstract From a theoretical puzzle to applications in cryptography and computer sci- ence: The factorization of prime numbers. In this paper we will introduce a historical retrospect by observing different methods of factorizing primes and we will introduce a theoretical approach to the General Number Field Sieve building from a foundation in Algebra and Number Theory. We will in this exclude most considerations of efficiency and practical im- plementation, and instead focus on the mathematical background. In this paper we will introduce the theory of algebraic number fields and Dedekind domains and their importance in understanding the General Number Field Sieve before continuing to explain, step by step, the inner workings of the General Number Field Sieve. Page 1 of 73 Contents Abstract 1 Table of contents 2 1 Introduction 3 2 Prime Numbers and the Algebra of Modern Cryptography 5 2.1 Preliminary Algebra and Number Theory .
    [Show full text]