DOCSLIB.ORG

Numbers, Groups and Cryptography Gordan Savin

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Numbers, Groups and Cryptography Gordan Savin Numbers, Groups and Cryptography Gordan Savin Contents Chapter 1. Euclidean Algorithm 5 1. Euclidean Algorithm 5 2. Fundamental Theorem of Arithmetic 9 3. Uniqueness of Factorization 14 4. Efficiency of the Euclidean Algorithm 16 Chapter 2. Groups and Arithmetic 21 1. Groups 21 2. Congruences 25 3. Modular arithmetic 28 4. Theorem of Lagrange 31 5. Chinese Remainder Theorem 34 Chapter 3. Rings and Fields 41 1. Fields and Wilson's Theorem 41 2. Field Characteristic and Frobenius 45 3. Quadratic Numbers 48 Chapter 4. Primes 53 1. Infinitude of primes 53 2. Primes in progression 56 3. Perfect Numbers and Mersenne Primes 59 4. The Lucas Lehmer test 62 Chapter 5. Roots 67 1. Roots 67 2. Property of the Euler Function 70 3. Primitive roots 71 4. Discrete logarithm 75 5. Cyclotomic Polynomials 77 Chapter 6. Quadratic Reciprocity 81 1. Squares modulo p 81 2. Fields of order p2 85 3. When is 2 a square modulo p? 88 4. Quadratic Reciprocity 91 Chapter 7. Applications of Quadratic Reciprocity 95 3 4 CONTENTS 1. Fermat primes 95 2. Quadratic Fields and the Circle Group 98 3. Lucas Lehmer test revisited 100 Chapter 8. Sums of two squares 107 1. Sums of two squares 107 2. Gaussian integers 111 3. Method of descent revisited 116 Chapter 9. Pell's Equations 119 1. Shape numbers and Induction 119 2. Square-Triangular Numbers and Pell Equation 122 3. Dirichlet's approximation 125 4. Pell Equation: classifying solutions 128 5. Continued Fractions 130 Chapter 10. Cryptography 135 1. Diffie-Hellman key exchange 135 2. RSA Code 138 3. ElGamal Code 142 Chapter 11. Primality testing 147 1. Miller Rabin test 147 2. p − 1 method 151 3. p + 1 method 154 4. Quadratic sieve 157 Chapter 12. Elliptic curves 161 1. Cubic curves 161 2. Degenerate curves 165 3. Curves modulo p 170 4. The curve y2 = x3 + bx 174 Chapter 13. Factoring and testing using elliptic curves 177 1. Lenstra's factoring method 177 2. Degenerate curves modulo p 181 3. Elliptic curve test for Mersenne primes 184 CHAPTER 1 Euclidean Algorithm 1. Euclidean Algorithm Euclid was a Greek mathematician who lived in Alexandria around 300 B.C. He devised a clever and fast algorithm to find the greatest common divisor of two integers. In modern language, the Euclidean Algorithm is the division with remainder. Recall that dividing two positive integers a; b means finding a positive integer q, called a quotient, such that a = qb + r with 0 ≤ r < b. The number r is a remainder of the division. If r = 0 then we say that b divides a, and write bja. For example, 3 j 6 whereas 4 - 6: Dividing two integers is probably the most difficult of the four standard binary operations. However, some of the most fundamental properties, such as the uniqueness of factorization, are based on the Euclidean Algorithm. But how is this related to finding the Greatest Common Divisor (or simply gcd) of two integers m and n? If factorizations of m and n into prime factors are known, then it is easy to figure out what the gcd is. Consider, for example, m = 756 and n = 360. Then 756 = 22 · 33 · 7 and 360 = 23 · 32 · 5: We see that 2 and 3 are the only primes appearing in factorizations of both numbers. Moreover, 22 and 32 are the greatest powers of 2 and 3, respectively, which divide both 756 and 360. It follows that gcd(756; 360) = 22 · 32 = 36: There are two issues here, however. Firstly, we have secretly assumed the uniqueness of factorization. The second issue, also very important, is that factoring into primes is a very difficult process, in general. A much better way was discovered by Euclid. Assume, for example, that we want to find the greatest common divisor of 60 and 22. Subtract 60 − 22 = 38. Notice that any number dividing 60 and 22 also divides 22 and 38. Conversely, any number that divides 22 and 38 also divides 22 + 38 = 60 and 22. Thus, instead of looking for common divisors of 60 and 22, we can look at common divisors of 22 and 38 instead. Since the later pair of numbers is smaller, the 5 6 1. EUCLIDEAN ALGORITHM problem of finding the greatest common divisor of 60 and 22 has just become easier, and we have accomplished that without ever dividing or multiplying two numbers. In fact, we can do even better. Instead of subtracting 22 from 60 once, we can subtract it twice, to get 60 − 2 × 22 = 16. Thus we get that gcd(60; 22) = gcd(22; 16): Of course, we do not stop here. Since 22 − 16 = 6, it follows that gcd(22; 16) = gcd(16; 6) and so on. At every step we replace a pair (a; b) by a pair (b; r) where r is the remainder of the division of a by b. The whole division process in this case is given here: 60 = 2 · 22 + 16 22 = 1 · 16 + 6 16 = 2 · 6 + 4 6 = 1 · 4 + 2 4 = 2 · 2 + 0 This shows that gcd(60; 22) = gcd(22; 16) = ··· = gcd(4; 2) = 2: The last statement is obvious since 2 divides 4. In general, starting with a pair of numbers a and b we use the division algorithm to generate a sequence of numbers (b > r1 > r2 > : : :) as follows. First, we divide a by b: a = q1b + r1; then divide b by r1 , and so on... b = q2r1 + r2 r1 = q3r2 + r3 . rn−2 = qnrn−1 + rn rn−1 = qn+1rn + 0: This process stops when the remainder is 0. Since b > r1 > r2 ::: it stops in less than b steps. We claim that the last non-zero remainder rn is equal to the greatest common divisor of a and b. In order to verify this, notice that the first equation a = qb + r1 implies that any common divisor of b and r1 also divides a. Likewise, if we rewrite the first equation as a − qb = r1, it is clear that any common divisor of a and b also divides r1. This shows that gcd(a; b) = gcd(b; r1). Arguing in this fashion, we see that gcd(a; b) = gcd(b; r1) = ::: = gcd(rn−1; rn) = rn 1. EUCLIDEAN ALGORITHM 7 where, of course, the last statement is obvious since rn divides rn−1. We can summarize what we have discovered: Euclidean Algorithm gives an effective way to compute the greatest com- mon divisor of two integers. Moreover, the algorithm does not rely on the uniqueness of factorization. The Euclidean Algorithm can be viewed as a special case of the Contin- ued Fraction Algorithm. We need some notation before proceeding to the definition of the algorithm. If α is a real number then letp [α] denote the greatest integer less than or equal to αp. For example, since 2 = 1:4 ::: the greatest integer less than or equal to 2 is 1: p [ 2] = 1: Note that [α] = α if α is an integer. The Continued Fraction Algorithm is defined as follows: (1) Let α > 1, and put β = α − [α]. (2) If β = 0 stop, else put α1 = 1/β and go to (1). In order to illustrate the relation between the two algorithms, let us work out the case when α = 60=22. Then the Continued Fraction Algorithm generates the following numerical data. i [αi] βi αi+1 0 2 16=22 22=16 1 1 6=16 16=6 2 2 4=6 6=4 3 1 2=4 4=2 4 2 0 STOP Here, of course, we put α0 = α and β0 = β. As it can be seen from the table, the Continued Fraction Algorithm generates the same numbers as the Euclidean Algorithm starting with the pair (60; 22). More precisely, if α = a=b and b > r1 > r2 ::: are non-negative integers generated by the Euclidean Algorithm applied to the pair (a; b), then r1 r2 r3 β = ; β1 = ; β2 = ::: b r1 r2 and [α] = q1; [α1] = q2; [α2] = q3 ::: Since rn+1 = 0 for some n, βn = 0 and the algorithm stops for every rational number (fraction). The name (continued fraction) comes from the fact the process can be recorded as 60 16 1 = 2 + = 2 + 6 = ::: 22 22 1 + 22 8 1. EUCLIDEAN ALGORITHM 1 ::: = 2 + 1 : 1 + 1 2+ 1 1+ 2 The continued fraction, in turn, gives a series of - so called - partial conver- gents: 1 1 1 1 2; 2 + ; 2 + 1 ; 2 + 1 and 2 + 1 : 1 1 + 2 1 + 1 1 + 1 2+ 1 2+ 1 1+ 2 An easy calculation gives that these five partial convergents are equal to, respectively, 8 11 30 60 2; 3; ; and = : 3 4 11 22 Ap rather different phenomenon occurs when we apply the algorithm to α = 2. Then going through the loop for the first time yields p p p p (1) α = 2; β = 2 − [ 2] = 2 − 1. (2) 1 1 p α1 = = p = 2 + 1: β 2 − 1 Going through the loop for the second time yields p p p p (1) α1 = 2 + 1; β1 = 2 + 1 − [ 2 + 1] = 2 − 1. (2) 1 1 p α2 = = p = 2 + 1: β1 2 − 1 p Thus the second output α2 = 2 + 1 is the same as the first output of the algorithm. Of course, the third output will be the same, and so on: p α1 = α2 = α3 = ::: = 2 + 1: We see that the Continued Fractionp Algorithm is stuck in the loop in thisp case.
Load more

Recommended publications
  • The Group of Classes of Congruent Quadratic Integers with Respect to Any Composite Ideal Modulus Whatever
    THE GROUPOF CLASSESOF CONGRUENTQUADRATIC INTEGERS WITH RESPECTTO A COMPOSITEIDEAL MODULUS* BY ARTHUR RANUM Introduction. If in the ordinary theory of rational numbers we consider a composite integer m as modulus, and if from among the classes of congruent integers with respect to that modulus we select those which are prime to the modulus, they form a well-known multiplicative group, which has been called by Weber (Algebra, vol. 2, 2d edition, p. 60), the most important example of a finite abelian group. In the more general theory of numbers in an algebraic field we may in a corre- sponding manner take as modulus a composite ideal, which includes as a special case a composite principal ideal, that is, an integer in the field, and if we regard all those integers of the field which are congruent to one another with respect to the modulus as forming a class, and if we select those classes whose integers are prime to the modulus, they also will form a finite abelian group f under multiplication. The investigation of the nature of this group is the object of the present paper. I shall confine my attention, however, to a quadratic number-field, and shall determine the structure of the group of classes of congruent quadratic integers with respect to any composite ideal modulus whatever. Several distinct cases arise depending on the nature of the prime ideal factors of the modulus ; for every case I shall find a complete system of independent generators of the group. Exactly as in the simpler theory of rational numbers it will appear that the solution of the problem depends essentially on the case in which the modulus is a prime-power ideal, that is, a power of a prime ideal.
    [Show full text]
  • Small-Span Hermitian Matrices Over Quadratic Integer Rings
    MATHEMATICS OF COMPUTATION Volume 84, Number 291, January 2015, Pages 409–424 S 0025-5718(2014)02836-6 Article electronically published on May 19, 2014 SMALL-SPAN HERMITIAN MATRICES OVER QUADRATIC INTEGER RINGS GARY GREAVES Abstract. A totally-real polynomial in Z[x] with zeros α1 α2 ··· αd has span αd−α1. Building on the classification of all characteristic polynomials of integer symmetric matrices having small span (span less than 4), we obtain a classification of small-span polynomials that are the characteristic polynomial of a Hermitian matrix over some quadratic integer ring. Taking quadratic integer rings as our base, we obtain as characteristic polynomials some low- degree small-span polynomials that are not the characteristic (or minimal) polynomial of any integer symmetric matrix. 1. Introduction and preliminaries 1.1. Introduction. Let f be a totally-real monic polynomial of degree d having integer coefficients with zeros α1 α2 ··· αd.Thespan of f is defined to be αd − α1. In this article, for Hermitian matrices A over quadratic integer rings, we are interested in the characteristic polynomials χA(x):=det(xI − A)andthe minimal polynomials pA(x). We build on the work of McKee [17] who classified all integer symmetric matrices whose characteristic polynomials have span less than 4. Following McKee, we call a totally-real monic integer polynomial small-span if its span is less than 4 and a Hermitian matrix is called small-span if its characteristic polynomial is small-span. A set consisting of an algebraic integer and all of its (Galois) conjugates is called a conjugate set.
    [Show full text]
  • Algebraic Numbers, Algebraic Integers
    Algebraic Numbers, Algebraic Integers October 7, 2012 1 Reading Assignment: 1. Read [I-R]Chapter 13, section 1. 2. Suggested reading: Davenport IV sections 1,2. 2 Homework set due October 11 1. Let p be a prime number. Show that the polynomial f(X) = Xp−1 + Xp−2 + Xp−3 + ::: + 1 is irreducible over Q. Hint: Use Eisenstein's Criterion that you have proved in the Homework set due today; and find a change of variables, making just the right substitution for X to turn f(X) into a polynomial for which you can apply Eisenstein' criterion. 2. Show that any unit not equal to ±1 in the ring of integers of a real quadratic field is of infinite order in the group of units of that ring. 3. [I-R] Page 201, Exercises 4,5,7,10. 4. Solve Problems 1,2 in these notes. 3 Recall: Algebraic integers; rings of algebraic integers Definition 1 An algebraic integer is a root of a monic polynomial with rational integer coefficients. Proposition 1 Let θ 2 C be an algebraic integer. The minimal1 monic polynomial f(X) 2 Q[X] having θ as a root, has integral coefficients; that is, f(X) 2 Z[X] ⊂ Q[X]. 1i.e., minimal degree 1 Proof: This follows from the fact that a product of primitive polynomials is again primitive. Discuss. Define content. Note that this means that there is no ambiguity in the meaning of, say, quadratic algebraic integer. It means, equivalently, a quadratic number that is an algebraic integer, or number that satisfies a quadratic monic polynomial relation with integral coefficients, or: Corollary 1 A quadratic number is a quadratic integer if and only if its trace and norm are integers.
    [Show full text]
  • Idempotent Factorizations of Singular $2\Times 2$ Matrices Over Quadratic
    IDEMPOTENT FACTORIZATIONS OF SINGULAR 2 2 × MATRICES OVER QUADRATIC INTEGER RINGS LAURA COSSU AND PAOLO ZANARDO Abstract. Let D be the ring of integers of a quadratic number field Q[√d]. We study the factorizations of 2 2 matrices over D into idem- × potent factors. When d < 0 there exist singular matrices that do not admit idempotent factorizations, due to results by Cohn (1965) and by the authors (2019). We mainly investigate the case d > 0. We employ Vaserˇste˘ın’s result (1972) that SL2(D) is generated by elementary ma- trices, to prove that any 2 2 matrix with either a null row or a null × column is a product of idempotents. As a consequence, every column- row matrix admits idempotent factorizations. Introduction In 1967 J. A. Erdos [10] proved that every square matrix with entries in a field that is singular (i.e., with zero determinant) admits idempotent factorizations, i.e., it can be written as a product of idempotent matrices. His work originated the problem of characterising integral domains R that satisfy the property IDn: every singular n n matrix with entries in R admits idempotent factorizations. We recall the× papers by Laffey [15] (1983), Hannah and O’Meara [13] (1989), Fountain [12] (1991). The related problem of factorizing invertible matrices into products of elementary factors has been a prominent subject of research over the years (see [3,5,16,17,19,23]). Following Cohn [5], we say that R satisfies property GEn if every invertible n n matrix over R can be written as a product of elementary matrices.
    [Show full text]
  • New York Journal of Mathematics Quadratic Integer Programming And
    New York Journal of Mathematics New York J. Math. 22 (2016) 907{932. Quadratic integer programming and the Slope Conjecture Stavros Garoufalidis and Roland van der Veen Abstract. The Slope Conjecture relates a quantum knot invariant, (the degree of the colored Jones polynomial of a knot) with a classical one (boundary slopes of incompressible surfaces in the knot complement). The degree of the colored Jones polynomial can be computed by a suitable (almost tight) state sum and the solution of a corresponding quadratic integer programming problem. We illustrate this principle for a 2-parameter family of 2-fusion knots. Combined with the results of Dunfield and the first author, this confirms the Slope Conjecture for the 2-fusion knots of one sector. Contents 1. Introduction 908 1.1. The Slope Conjecture 908 1.2. Boundary slopes 909 1.3. Jones slopes, state sums and quadratic integer programming 909 1.4. 2-fusion knots 910 1.5. Our results 911 2. The colored Jones polynomial of 2-fusion knots 913 2.1. A state sum for the colored Jones polynomial 913 2.2. The leading term of the building blocks 916 2.3. The leading term of the summand 917 3. Proof of Theorem 1.1 917 3.1. Case 1: m1; m2 ≥ 1 918 3.2. Case 2: m1 ≤ 0; m2 ≥ 1 919 3.3. Case 3: m1 ≤ 0; m2 ≤ −2 919 Received January 13, 2016. 2010 Mathematics Subject Classification. Primary 57N10. Secondary 57M25. Key words and phrases. knot, link, Jones polynomial, Jones slope, quasi-polynomial, pretzel knots, fusion, fusion number of a knot, polytopes, incompressible surfaces, slope, tropicalization, state sums, tight state sums, almost tight state sums, regular ideal octa- hedron, quadratic integer programming.
    [Show full text]
  • Fast Exponentiation Via Prime Finite Field Isomorphism
    Alexander Rostovtsev, St. Petersburg State Polytechnic University [email protected] Fast exponentiation via prime finite field isomorphism Raising of the fixed element of prime order group to arbitrary degree is the main op- eration specified by digital signature algorithms DSA, ECDSA. Fast exponentiation algorithms are proposed. Algorithms use number system with algebraic integer base 4 ±2 . Prime group order r can be factored r =pp in Euclidean ring [±2] by Pol- lard and Schnorr algorithm. Farther factorization of prime quadratic integer p=rr 4 in the ring [±2] can be done similarly. Finite field r is represented as quotient 4 ring [±r2]/(). Integer exponent k is reduced in corresponding quotient ring by minimization of absolute value of its norm. Algorithms can be used for fast exponen- tiation in arbitrary cyclic group if its order can be factored in corresponding number rings. If window size is 4 bits, this approach allows speeding-up 2.5 times elliptic curve digital signature verification comparatively to known methods with the same window size. Key words: fast exponentiation, cyclic group, algebraic integers, digital signature, elliptic curve. 1. Introduction Let G be cyclic group of prime computable order. Raising of the fixed element a Î G to arbitrary degree is the main operation specified by many public key cryptosystems, such as digital signature algorithms DSA, ECDSA [2]. In crypto- graphic applications prime group order is more then 2100. Group G can be given as subgroup of finite field [6], elliptic curve or Jacobian of hyperelliptic curve over finite field, class group of number field, etc.
    [Show full text]
  • A Computational Exploration of Gaussian and Eisenstein Moats
    A Computational Exploration of Gaussian and Eisenstein Moats By Philip P. West A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS IN SCIENCE (MATHEMATICS) at the CALIFORNIA STATE UNIVERSITY CHANNEL ISLANDS 20 14 copyright 20 14 Philip P. West ALL RIGHTS RESERVED APPROVED FOR THE MATHEMATICS PROGRAM Brian Sittinger, Advisor Date: 21 December 20 14 Ivona Grzegorzyck Date: 12/2/14 Ronald Rieger Date: 12/8/14 APPROVED FOR THE UNIVERSITY Doctor Gary A. Berg Date: 12/10/14 Non- Exclusive Distribution License In order for California State University Channel Islands (C S U C I) to reproduce, translate and distribute your submission worldwide through the C S U C I Institutional Repository, your agreement to the following terms is necessary. The authors retain any copyright currently on the item as well as the ability to submit the item to publishers or other repositories. By signing and submitting this license, you (the authors or copyright owner) grants to C S U C I the nonexclusive right to reproduce, translate (as defined below), and/ or distribute your submission (including the abstract) worldwide in print and electronic format and in any medium, including but not limited to audio or video. You agree that C S U C I may, without changing the content, translate the submission to any medium or format for the purpose of preservation. You also agree that C S U C I may keep more than one copy of this submission for purposes of security, backup and preservation. You represent that the submission is your original work, and that you have the right to grant the rights contained in this license.
    [Show full text]
  • Analysis of Fast Versions of the Euclid Algorithm Eda Cesaratto, Julien Clément, Benoît Daireaux, Loïck Lhote, Véronique Maume-Deschamps, Brigitte Vallée
    Analysis of fast versions of the euclid algorithm Eda Cesaratto, Julien Clément, Benoît Daireaux, Loïck Lhote, Véronique Maume-Deschamps, Brigitte Vallée To cite this version: Eda Cesaratto, Julien Clément, Benoît Daireaux, Loïck Lhote, Véronique Maume-Deschamps, et al.. Analysis of fast versions of the euclid algorithm. 2008. hal-00211424 HAL Id: hal-00211424 https://hal.archives-ouvertes.fr/hal-00211424 Preprint submitted on 21 Jan 2008 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. Analysis of fast versions of the Euclid Algorithm Eda Cesaratto ∗ Julien Cl´ement † Benoˆıt Daireaux‡ Lo¨ıck Lhote § V´eronique Maume-Deschamps ¶ Brigitte Vall´ee k To appear in the ANALCO’07 Proceedings Abstract gebra. Many gcd algorithms have been designed since There exist fast variants of the gcd algorithm which are Euclid. Most of them compute a sequence of remainders all based on principles due to Knuth and Sch¨onhage. On by successive divisions, which leads to algorithms with inputs of size n, these algorithms use a Divide and Con- a quadratic bit–complexity (in the worst-case and in the quer approach, perform FFT multiplications and stop average-case).
    [Show full text]
  • Optimality of the Width-$ W $ Non-Adjacent Form: General
    OPTIMALITY OF THE WIDTH-w NON-ADJACENT FORM: GENERAL CHARACTERISATION AND THE CASE OF IMAGINARY QUADRATIC BASES CLEMENS HEUBERGER AND DANIEL KRENN Abstract. Efficient scalar multiplication in Abelian groups (which is an im- portant operation in public key cryptography) can be performed using digital expansions. Apart from rational integer bases (double-and-add algorithm), imaginary quadratic integer bases are of interest for elliptic curve cryptogra- phy, because the Frobenius endomorphism fulfils a quadratic equation. One strategy for improving the efficiency is to increase the digit set (at the prize of additional precomputations). A common choice is the width-w non-adjacent form (w-NAF): each block of w consecutive digits contains at most one non- zero digit. Heuristically, this ensures a low weight, i.e. number of non-zero digits, which translates in few costly curve operations. This paper investigates the following question: Is the w-NAF-expansion optimal, where optimality means minimising the weight over all possible expansions with the same digit set? The main characterisation of optimality of w-NAFs can be formulated in the following more general setting: We consider an Abelian group together with an endomorphism (e.g., multiplication by a base element in a ring) and a finite digit set. We show that each group element has an optimal w-NAF- expansion if and only if this is the case for each sum of two expansions of weight 1. This leads both to an algorithmic criterion and to generic answers for various cases. Imaginary quadratic integers of trace at least 3 (in absolute value) have optimal w-NAFs for w 4.
    [Show full text]
  • Answers and Solutions
    Answers and Solutions Chapter 1 1.1(a). The point E can be obtained by folding the square along the bisector of angle BAC, so that AB is folded onto the diagonal with B landing on E. If F is the point where the bisector intersects B C, then B F folds to E F. Thus E F -1 AC and BF = EF. Since L.EFC = 45° = L.ECF, FE = EC. l.1(b). WC! = IBC! - IBFI = IAEI - ICEI = 21AEI - lAC! and IEC! = lAC! - IAEI = lAC! - IBC!. l.3(a). n Pn qn rn 1 1 1 1 2 3 2 3/2 = 1.5 3 7 5 7/5 = 1.4 4 17 12 17/12 = 1.416666 5 41 29 41/29 = 1.413793 1.3(c). Clearly, rn > 1 for each n, so that from (b), rn+1 - rn and rn - rn-I have opposite signs and 1 Irn+l - rnl < 41rn - rn-II· Since r2 > 1 = rl, it follows that rl < r3 < r2. Suppose as an induction hypothesis that rl < r3 < ... < r2m-1 < r2m < ... < r2. Then 0 < r2m - r2m+1 < r2m - r2m-1 and 0 < r2m+2 - r2m+1 < r2m - r2m+J. so that r2m-1 < r2m+1 < r2m+2 < r2m. Let k, 1 be any positive integers. Then, for m > k, I, r2k+1 < r2m-1 < r2m < r21. l.3(d). Leta be the least upper bound (i.e., the smallest number at least as great as all) of the numbers in the set {rl, r3, r5, ... , r2k+l, ... }. Then r2m-1 ::::: a ::::: r2m 139 140 Answers and Solutions .....-_______---,D \ \ \ \ \ \ \ \ \ \ BL..---...L..-----~ F FIGURE 1.3.
    [Show full text]
  • Orders in Quadratic Imaginary Fields of Small Class Number
    ORDERS IN QUADRATIC IMAGINARY FIELDS OF SMALL CLASS NUMBER JANIS KLAISE Abstract. In 2003 M. Watkins [Wat03] solved the Gauss Class Number Prob- lem for class numbers up to 100: given a class number h, what is the complete list of quadratic imaginary fields K with this class number? In this project we extend the problem by including non-maximal orders - subrings O of the ring of integers OK that are also Z-modules of rank 2 admiting a Q-basis of K. These orders are more complicated than OK since they need not be Dedekind domains. The main theoretical part is the class number formula linking the class numbers of O and OK . We give a complete proof of this and use it together with the results of Watkins to find the orders of class number h = 1, 2, 3 by solving the class number formula in a Diophantine fashion. We then describe a different algorithm that was suggested and implemented dur- ing the writing of this project and succesfully finds all orders of class number up to 100. Finally, we motivate the problem by its application to j-invariants in complex multiplication. Contents 1. Introduction 1 2. The Class Number Formula 4 3. Example Calculations. 13 4. An Algorithm 18 5. An Application to Complex Multiplication 20 References 23 1. Introduction Gauss in his Disquisitiones Arithmeticae [Gau86, §303-304] came up with several conjectures about imaginary quadratic fields (in the language of discriminants of integral binary quadratic forms). In particular, he gave lists of discriminants cor- responding to imaginary quadratic fields with given small class number and be- lieved them to be complete without proof.
    [Show full text]
  • MTH6128 Number Theory 2 Algebraic Numbers
    MTH6128 Number Theory Notes 2 Spring 2020 \The only way to learn Mathematics is to do Mathematics." Paul Halmos 2 Algebraic numbers 2.1 Algebraic numbers and algebraic integers Pythagoras and his school discovered that the square root of 2 is not a rational number. (You saw the proof of this in Introduction to Algebra.) However, it is an easy number to describe geometrically: it is the ratio of the diagonal of a square to its side. The number π has a more complicated geometric description: it is the ratio of the circumference of a circle to its diameter, but there is no simple method to construct a straight line which is equal to the circumference of a given circle. (We know now, for example, that such a line cannot be constructed with the traditional geometric instruments of \ruler and compass".) We make a distinction between algebraic numbers (which are roots of polyno- mials with rational coefficients) and transcendental numbers (which are not): Definition Let α be a complex number. Then: • α is an algebraic number if there is a non-zero polynomial f(x) with rati- onal coefficients such that f(α) = 0; • α is a transcendental number if α is not an algebraic number. Moreover, • α is an algebraic integer if there is a non-zero monic polynomial f(x) with integer coefficients such that f(α) = 0. (A polynomial is monic if the coefficient of the highest power of x is equal to 1, n n−1 i.e., it is of the form f(x) = x +an−1x +:::+a1x+a0, with a0; : : : ; an−1 2 Z.) 1 Note that there are two extra conditions in the definition of an algebraic inte- ger.
    [Show full text]