DOCSLIB.ORG

Quadratic Reciprocity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quadratic Reciprocity QUADRATIC RECIPROCITY LEO GOLDMAKHER Quadratic Reciprocity is arguably the most important theorem taught in an elementary number theory course. Since Gauss’ original 1796 proof (by induction!) appeared, more than 100 different proofs have been discovered. Here I present one proof which is not particularly well-known, due to George Rousseau [1]. (The proof was rediscovered more recently by high-schooler Tim Kunisky.) Although not the shortest proof, it is the easiest to remember of all the elementary proofs I have encountered. In particular, it does not rely on Gauss’ Lemma, or lattice counting, or Gauss sums; the only ingredients used in the proof are the Chinese Remainder Theorem, Wilson’s Theorem, and Euler’s Criterion. I’ll recall these, and the statement of Quadratic Reciprocity, in the first section. In the second section, I’ll explain Rousseau’s proof. In the third and final section, I discuss the Jacobi symbol and the supplement to Quadratic Reciprocity. 1. PRELIMINARIES × Recall that Zn is defined to be the set of all integers between 1 and n which are relatively prime × to n. Actually, an element k 2 Zn doesn’t merely refer to the number k, but to the arithmetic × progression k + nZ. It will be convenient to refer to elements of Zn using the name of one of the terms in the corresponding arithmetic progression. For example, strictly speaking 9 isn’t an × × element of Z5 , but we will allow the symbol 9 as an alternative name for the element 4 2 Z5 . a Recall that the Legendre symbol, denoted p , measures whether or not a is a square (mod p). More precisely, it is defined as follows: for any integer a and any odd prime p, 8 0 if a ≡ 0 (mod p) a <> := 1 if 9x 2 × such that x2 ≡ a (mod p) p Zp :>−1 otherwise. 1 2 a Thus, 3 is 1 and 3 is −1. If p is 1 (i.e. a is a square (mod p)), we say a is a quadratic a residue modulo p. If p is −1, we say a is a quadratic non-residue. The following result, an easy consequence of the existence of a primitive root modulo p, is a very useful tool in studying the Legendre symbol. Theorem (Euler’s Criterion). Given any odd prime p and any a 2 Z, we have a ≡ a(p−1)=2 (mod p): p Thus, for example, −1 ≡ (−1)3 (mod 7); 7 whence −1 is not a square (mod 7). This example generalizes quite nicely: Corollary. Given an odd prime p, we have ( −1 1 if p ≡ 1 (mod 4) = p −1 if p ≡ −1 (mod 4): a Are there similar general formulas for p for other choices of a? Some experimentation will convince the reader that the answer is yes; however, it’s not at all obvious how to write down a master formula for the general a and p. The Law of Quadratic Reciprocity solves this problem in the case that a is an odd prime: Theorem (Quadratic Reciprocity). Given distinct odd primes p and q. Then pq p−1 · q−1 = (−1) 2 2 : q p 3 For example, this allows us to figure out a formula for p : by QR, we have 3 p−1 p = (−1) 2 : p 3 p p−1 2 Since 3 is completely determined by the reduction of p (mod 3), and (−1) is determined by the reduction of p (mod 4), we deduce the following formula: ( 3 1 if p ≡ ±1 (mod 12) = p −1 if p ≡ ±5 (mod 12): 5 7 q We can use QR to obtain similar formulas for p , or p , or more generally for p for any odd 2 15 prime q. But what about for other choices of top entry? Is there a formula for p , or for p ? As we shall see (in the third section), just by playing around with QR a bit one can derive formulas a for all of these, and more generally, for p for any fixed integer a. Our proof of QR relies crucially on the following result: Theorem (Chinese Remainder Theorem). Given positive relatively prime integers m; n. The fol- lowing map is a bijection: × × × σ : Zmn −! Zm × Zn k 7−! (k; k) × × × In other words, for all (a; b) 2 Zm × Zn there exists a unique k 2 Zmn such that σ(k) = (a; b). × I strongly encourage the reader to compute σ(k) for all k 2 Z15, with m = 3 and n = 5, to get a sense of how this works. For example, with these choices of m and n, we have σ(7) = (1; 2). We are now in a position to prove Quadratic Reciprocity, which we do in the next section. In the third and final section, we discuss extensions of QR. 2. PROOF Consider the sets n q o n pq o A := (a; b) 2 × × × : 1 ≤ b < and K := k 2 × : 1 ≤ k < : Zp Zq 2 Zpq 2 × × × A is half of Zp × Zq , while K is half of Zpq. Our first move is to adapt the Chinese Remainder Theorem to compare these two sets. Lemma 1. For each (a; b) 2 A, there exists a unique k 2 K such that σ(k) = ±(a; b). I strongly encourage the reader to experiment with the case p = 3; q = 5 to get a feel for this result. Lemma1 implies that Y Y Y (a; b) = ±σ(k) = ± (k; k): (a;b)2A k2K k2K In other words, there exists = ±1 such that the following congruences simultaneously hold: Y Y Y a ≡ k (mod p) (1) × a2Zp b<q=2 k<pq=2 (k;pq)=1 and Y Y Y b ≡ k (mod q): (2) × a2Zp b<q=2 k<pq=2 (k;pq)=1 Evaluating these products isn’t too difficult, as we shall see (after finishing the proof of QR). We will show: p−1 q−1 Lemma 2. Let P := 2 and Q := 2 . Then Y Y Y (−1)Q a ≡ (−1)Q (mod p) k ≡ (mod p) qP × a2Zp b<q=2 k<pq=2 (k;pq)=1 Y Y Y (−1)P b ≡ (−1)P (−1)PQ (mod q) k ≡ (mod q) pQ × a2Zp b<q=2 k<pq=2 (k;pq)=1 By Euler’s Criterion, q p qP ≡ (mod p) and pQ ≡ (mod q): p q q Plugging this into Lemma2 and equation (1) shows that = p . Substituting this into congruence (2) and applying Lemma2 yields p q (−1)PQ = : q p p Multiplying both sides by q concludes the proof. QED The above is an overview of the proof of Quadratic Reciprocity. To complete the proof, it suffices to prove the two lemmas. I leave the first lemma as an exercise to the reader. The second lemma consists of four congruences, which we prove now. Let’s start with the easiest of the four. We have Y Y Y a = aQ = (p − 1)!Q: × × a2Zp b<q=2 a2Zp By Wilson’s theorem, (p − 1)! ≡ −1 (mod p), whence Y Y a ≡ (−1)Q (mod p) × a2Zp b<q=2 as claimed. Next, we have Y Y P b = (Q!)p−1 = (Q!2)P = Q! · (−1)(−2) ··· (−Q) · (−1)Q × a2Zp b<q=2 P ≡ (q − 1)!(−1)Q (mod q) ≡ (−1)P (−1)PQ (mod q): Finally, we will compute Y k (mod p): (3) k<pq=2 (k;pq)=1 We’re multiplying together all the integers between 1 and pq=2 which are not multiples of p or q. To gain some intuition, we start by writing down a table of all integers between 1 and pq=2: 1 2 3 ··· p p + 1 p + 2 p + 3 ··· 2p 2p + 1 2p + 2 2p + 3 ··· 3p . (Q − 1)p + 1 (Q − 1)p + 2 (Q − 1)p + 3 ··· Qp pq−1 Qp + 1 Qp + 2 Qp + 3 ··· 2 This is a (Q × p) table, plus an incomplete (Q + 1)st row. (Prove this!) To compute the product (3) we need to remove all multiples of p and q from the table, and multiply the rest of the elements together. The multiples of p are easy to remove. Once that’s done, here’s how we proceed: • each complete row contributes (p − 1)! (mod p) to the product, and there are Q complete rows; • the incomplete row contributes P ! (mod p) to the product (why?); and • there are precisely P multiples of q in the table, namely q; 2q; 3q; : : : ; P q. (Prove this!) Thus, we find Y (p − 1)!Q · P ! k ≡ (mod p) (q)(2q)(3q) ··· (P q) k≤pq=2 (k;pq)=1 (−1)Q · P ! ≡ (mod p) P ! · qP (−1)Q ≡ (mod p): qP The fourth congruence, which is the evaluation of the same product as above but modulo q, follows from the above computation by switching the roles of p and q. 3. EXTENSIONS 3 10 Quadratic Reciprocity allows us to calculate Legendre symbols like 47 . But what about 47 ? In this section, we’ll prove two results which will allow us to evaluate such symbols as well. The first is a multiplicative property of the Legendre symbol: Proposition 3. For any two integers a; b and any odd prime p, ab a b = : p p p Proof. This is an immediate consequence of Euler’s Criterion. 10 2 5 Thus, 47 = 47 47 . The second factor can now be evaluated by Quadratic Reciprocity, so the 2 only remaining question is a formula for p . We will prove: Theorem 4.
Load more

Recommended publications
  • Quadratic Reciprocity and Computing Modular Square Roots.Pdf
    12 Quadratic reciprocity and computing modular square roots In §2.8, we initiated an investigation of quadratic residues. This chapter continues this investigation. Recall that an integer a is called a quadratic residue modulo a positive integer n if gcd(a, n) = 1 and a ≡ b2 (mod n) for some integer b. First, we derive the famous law of quadratic reciprocity. This law, while his- torically important for reasons of pure mathematical interest, also has important computational applications, including a fast algorithm for testing if an integer is a quadratic residue modulo a prime. Second, we investigate the problem of computing modular square roots: given a quadratic residue a modulo n, compute an integer b such that a ≡ b2 (mod n). As we will see, there are efficient probabilistic algorithms for this problem when n is prime, and more generally, when the factorization of n into primes is known. 12.1 The Legendre symbol For an odd prime p and an integer a with gcd(a, p) = 1, the Legendre symbol (a j p) is defined to be 1 if a is a quadratic residue modulo p, and −1 otherwise. For completeness, one defines (a j p) = 0 if p j a. The following theorem summarizes the essential properties of the Legendre symbol. Theorem 12.1. Let p be an odd prime, and let a, b 2 Z. Then we have: (i) (a j p) ≡ a(p−1)=2 (mod p); in particular, (−1 j p) = (−1)(p−1)=2; (ii) (a j p)(b j p) = (ab j p); (iii) a ≡ b (mod p) implies (a j p) = (b j p); 2 (iv) (2 j p) = (−1)(p −1)=8; p−1 q−1 (v) if q is an odd prime, then (p j q) = (−1) 2 2 (q j p).
    [Show full text]
  • Appendices A. Quadratic Reciprocity Via Gauss Sums
    1 Appendices We collect some results that might be covered in a first course in algebraic number theory. A. Quadratic Reciprocity Via Gauss Sums A1. Introduction In this appendix, p is an odd prime unless otherwise specified. A quadratic equation 2 modulo p looks like ax + bx + c =0inFp. Multiplying by 4a, we have 2 2ax + b ≡ b2 − 4ac mod p Thus in studying quadratic equations mod p, it suffices to consider equations of the form x2 ≡ a mod p. If p|a we have the uninteresting equation x2 ≡ 0, hence x ≡ 0, mod p. Thus assume that p does not divide a. A2. Definition The Legendre symbol a χ(a)= p is given by 1ifa(p−1)/2 ≡ 1modp χ(a)= −1ifa(p−1)/2 ≡−1modp. If b = a(p−1)/2 then b2 = ap−1 ≡ 1modp,sob ≡±1modp and χ is well-defined. Thus χ(a) ≡ a(p−1)/2 mod p. A3. Theorem a The Legendre symbol ( p ) is 1 if and only if a is a quadratic residue (from now on abbre- viated QR) mod p. Proof.Ifa ≡ x2 mod p then a(p−1)/2 ≡ xp−1 ≡ 1modp. (Note that if p divides x then p divides a, a contradiction.) Conversely, suppose a(p−1)/2 ≡ 1modp.Ifg is a primitive root mod p, then a ≡ gr mod p for some r. Therefore a(p−1)/2 ≡ gr(p−1)/2 ≡ 1modp, so p − 1 divides r(p − 1)/2, hence r/2 is an integer. But then (gr/2)2 = gr ≡ a mod p, and a isaQRmodp.
    [Show full text]
  • QUADRATIC RECIPROCITY Abstract. the Goals of This Project Are to Have
    QUADRATIC RECIPROCITY JORDAN SCHETTLER Abstract. The goals of this project are to have the reader(s) gain an appreciation for the usefulness of Legendre symbols and ultimately recreate Eisenstein's slick proof of Gauss's Theorema Aureum of quadratic reciprocity. 1. Quadratic Residues and Legendre Symbols Definition 0.1. Let m; n 2 Z with (m; n) = 1 (recall: the gcd (m; n) is the nonnegative generator of the ideal mZ + nZ). Then m is called a quadratic residue mod n if m ≡ x2 (mod n) for some x 2 Z, and m is called a quadratic nonresidue mod n otherwise. Prove the following remark by considering the kernel and image of the map x 7! x2 on the group of units (Z=nZ)× = fm + nZ :(m; n) = 1g. Remark 1. For 2 < n 2 N the set fm+nZ : m is a quadratic residue mod ng is a subgroup of the group of units of order ≤ '(n)=2 where '(n) = #(Z=nZ)× is the Euler totient function. If n = p is an odd prime, then the order of this group is equal to '(p)=2 = (p − 1)=2, so the equivalence classes of all quadratic nonresidues form a coset of this group. Definition 1.1. Let p be an odd prime and let n 2 Z. The Legendre symbol (n=p) is defined as 8 1 if n is a quadratic residue mod p n < = −1 if n is a quadratic nonresidue mod p p : 0 if pjn: The law of quadratic reciprocity (the main theorem in this project) gives a precise relation- ship between the \reciprocal" Legendre symbols (p=q) and (q=p) where p; q are distinct odd primes.
    [Show full text]
  • Quadratic Reciprocity Laws*
    JOURNAL OF NUMBER THEORY 4, 78-97 (1972) Quadratic Reciprocity Laws* WINFRIED SCHARLAU Mathematical Institute, University of Miinster, 44 Miinster, Germany Communicated by H. Zassenhaus Received May 26, 1970 Quadratic reciprocity laws for the rationals and rational function fields are proved. An elementary proof for Hilbert’s reciprocity law is given. Hilbert’s reciprocity law is extended to certain algebraic function fields. This paper is concerned with reciprocity laws for quadratic forms over algebraic number fields and algebraic function fields in one variable. This is a very classical subject. The oldest and best known example of a theorem of this kind is, of course, the Gauss reciprocity law which, in Hilbert’s formulation, says the following: If (a, b) is a quaternion algebra over the field of rational numbers Q, then the number of prime spots of Q, where (a, b) does not split, is finite and even. This can be regarded as a theorem about quadratic forms because the quaternion algebras (a, b) correspond l-l to the quadratic forms (1, --a, -b, ub), and (a, b) is split if and only if (1, --a, -b, ub) is isotropic. This formulation of the Gauss reciprocity law suggests immediately generalizations in two different directions: (1) Replace the quaternion forms (1, --a, -b, ub) by arbitrary quadratic forms. (2) Replace Q by an algebraic number field or an algebraic function field in one variable (possibly with arbitrary constant field). While some results are known concerning (l), it seems that the situation has never been investigated thoroughly, not even for the case of the rational numbers.
    [Show full text]
  • Selmer Groups and Quadratic Reciprocity 3
    SELMER GROUPS AND QUADRATIC RECIPROCITY FRANZ LEMMERMEYER Abstract. In this article we study the 2-Selmer groups of number fields F as well as some related groups, and present connections to the quadratic reci- procity law in F . Let F be a number field; elements in F × that are ideal squares were called sin- gular numbers in the classical literature. They were studied in connection with explicit reciprocity laws, the construction of class fields, or the solution of em- bedding problems by mathematicians like Kummer, Hilbert, Furtw¨angler, Hecke, Takagi, Shafarevich and many others. Recently, the groups of singular numbers in F were christened Selmer groups by H. Cohen [4] because of an analogy with the Selmer groups in the theory of elliptic curves (look at the exact sequence (2.2) and recall that, under the analogy between number fields and elliptic curves, units correspond to rational points, and class groups to Tate-Shafarevich groups). In this article we will present the theory of 2-Selmer groups in modern language, and give direct proofs based on class field theory. Most of the results given here can be found in 61ff of Hecke’s book [11]; they had been obtained by Hilbert and Furtw¨angler in the§§ roundabout way typical for early class field theory, and were used for proving explicit reciprocity laws. Hecke, on the other hand, first proved (a large part of) the quadratic reciprocity law in number fields using his generalized Gauss sums (see [3] and [24]), and then derived the existence of quadratic class fields (which essentially is just the calculation of the order of a certain Selmer group) from the reciprocity law.
    [Show full text]
  • 4 Patterns in Prime Numbers: the Quadratic Reciprocity Law
    4 Patterns in Prime Numbers: The Quadratic Reciprocity Law 4.1 Introduction The ancient Greek philosopher Empedocles (c. 495–c. 435 b.c.e.) postulated that all known substances are composed of four basic elements: air, earth, fire, and water. Leucippus (fifth century b.c.e.) thought that these four were indecomposable. And Aristotle (384–322 b.c.e.) introduced four properties that characterize, in various combinations, these four elements: for example, fire possessed dryness and heat. The properties of compound substances were aggregates of these. This classical Greek concept of an element was upheld for almost two thousand years. But by the end of the nineteenth century, 83 chem- ical elements were known to exist, and these formed the basic building blocks of more complex substances. The European chemists Dmitry I. Mendeleyev (1834–1907) and Julius L. Meyer (1830–1895) arranged the elements approx- imately in the order of increasing atomic weight (now known to be the order of increasing atomic numbers), which exhibited a periodic recurrence of their chemical properties [205, 248]. This pattern in properties became known as the periodic law of chemical elements, and the arrangement as the periodic table. The periodic table is now at the center of every introductory chemistry course, and was a major breakthrough into the laws governing elements, the basic building blocks of all chemical compounds in the universe (Exercise 4.1). The prime numbers can be considered numerical analogues of the chemical elements. Recall that these are the numbers that are multiplicatively indecom- posable, i.e., divisible only by 1 and by themselves.
    [Show full text]
  • Notes on Mod P Arithmetic, Group Theory and Cryptography Chapter One of an Invitation to Modern Number Theory
    Notes on Mod p Arithmetic, Group Theory and Cryptography Chapter One of An Invitation to Modern Number Theory Steven J. Miller and Ramin Takloo-Bighash September 5, 2006 Contents 1 Mod p Arithmetic, Group Theory and Cryptography 1 1.1 Cryptography . 1 1.2 Efficient Algorithms . 3 1.2.1 Exponentiation . 3 1.2.2 Polynomial Evaluation (Horner’s Algorithm) . 4 1.2.3 Euclidean Algorithm . 4 1.2.4 Newton’s Method and Combinatorics . 6 1.3 Clock Arithmetic: Arithmetic Modulo n ................................ 10 1.4 Group Theory . 11 1.4.1 Definition . 12 1.4.2 Lagrange’s Theorem . 12 1.4.3 Fermat’s Little Theorem . 14 1.4.4 Structure of (Z=pZ)¤ ...................................... 15 1.5 RSA Revisited . 15 1.6 Eisenstein’s Proof of Quadratic Reciprocity . 17 1.6.1 Legendre Symbol . 17 1.6.2 The Proof of Quadratic Reciprocity . 18 1.6.3 Preliminaries . 19 1.6.4 Counting Lattice Points . 21 1 Abstract We introduce enough group theory and number theory to analyze in detail certain problems in cryptology. In the course of our investigations we comment on the importance of finding efficient algorithms for real world applica- tions. The notes below are from An Invitation to Modern Number Theory, published by Princeton University Press in 2006. For more on the book, see http://www.math.princeton.edu/mathlab/book/index.html The notes below are Chapter One of the book; as such, there are often references to other parts of the book. These references will look something like ?? in the text. If you want additional information on any of these references, please let me know.
    [Show full text]
  • Public-Key Cryptography
    http://dx.doi.org/10.1090/psapm/062 AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures published as a subseries of Proceedings of Symposia in Applied Mathematics Proceedings of Symposia in APPLIED MATHEMATICS Volume 62 Public-Key Cryptography American Mathematical Society Short Course January 13-14, 2003 Baltimore, Maryland Paul Garrett Daniel Lieman Editors ^tfEMAT , American Mathematical Society ^ Providence, Rhode Island ^VDED Editorial Board Mary Pugh Lenya Ryzhik Eitan Tadmor (Chair) LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE PUBLIC-KEY CRYPTOGRAPHY HELD IN BALTIMORE, MARYLAND JANUARY 13-14, 2003 The AMS Short Course Series is sponsored by the Society's Program Committee for National Meetings. The series is under the direction of the Short Course Subcommittee of the Program Committee for National Meetings. 2000 Mathematics Subject Classification. Primary 54C40, 14E20, 14G50, 11G20, 11T71, HYxx, 94Axx, 46E25, 20C20. Library of Congress Cataloging-in-Publication Data Public-key cryptography / Paul Garrett, Daniel Lieman, editors. p. cm. — (Proceedings of symposia in applied mathematics ; v. 62) Papers from a conference held at the 2003 Baltimore meeting of the American Mathematical Society. Includes bibliographical references and index. ISBN 0-8218-3365-0 (alk. paper) 1. Computers—Access control-—Congresses. 2. Public key cryptography—Congresses. I. Garrett, Paul, 1952— II. Lieman, Daniel, 1965- III. American Mathematical Society. IV. Series. QA76.9.A25P82 2005 005.8'2—dc22 2005048178 Copying and reprinting. Material in this book may be reproduced by any means for edu• cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg• ment of the source is given.
    [Show full text]
  • Primes and Quadratic Reciprocity
    PRIMES AND QUADRATIC RECIPROCITY ANGELICA WONG Abstract. We discuss number theory with the ultimate goal of understanding quadratic reciprocity. We begin by discussing Fermat's Little Theorem, the Chinese Remainder Theorem, and Carmichael numbers. Then we define the Legendre symbol and prove Gauss's Lemma. Finally, using Gauss's Lemma we prove the Law of Quadratic Reciprocity. 1. Introduction Prime numbers are especially important for random number generators, making them useful in many algorithms. The Fermat Test uses Fermat's Little Theorem to test for primality. Although the test is not guaranteed to work, it is still a useful starting point because of its simplicity and efficiency. An integer is called a quadratic residue modulo p if it is congruent to a perfect square modulo p. The Legendre symbol, or quadratic character, tells us whether an integer is a quadratic residue or not modulo a prime p. The Legendre symbol has useful properties, such as multiplicativity, which can shorten many calculations. The Law of Quadratic Reciprocity tells us that for primes p and q, the quadratic character of p modulo q is the same as the quadratic character of q modulo p unless both p and q are of the form 4k + 3. In Section 2, we discuss interesting facts about primes and \fake" primes (pseu- doprimes and Carmichael numbers). First, we prove Fermat's Little Theorem, then show that there are infinitely many primes and infinitely many primes congruent to 1 modulo 4. We also present the Chinese Remainder Theorem and using both it and Fermat's Little Theorem, we give a necessary and sufficient condition for a number to be a Carmichael number.
    [Show full text]
  • Quadratic Residues (Week 3-Monday)
    QUADRATIC RESIDUES (WEEK 3-MONDAY) We studied the equation ax b mod m and solved it when gcd(a, m)=1. I left it as an exercise to investigate what happens ⌘ when gcd(a, m) = 1 [Hint: There are no solutions to ax 1 mod m]. Thus, we have completely understood the equation 6 ⌘ ax b mod m. Then we took it one step further and found simultaneous solutions to a set of linear equations. The next natural ⌘ step is to consider quadratic congruences. Example 1 Find the solutions of the congruence 15x2 + 19x 5 mod 11. ⌘ A We need to find x such that 15x2 + 19x 5 4x2 + 8x 5 (2x 1)(2x + 5) 0 mod 11. Since 11 is prime, this happens − ⌘ − ⌘ − ⌘ iff 2x 1 0 mod 11 or 2x + 5 0 mod 11. Now you can finish this problem by using the techniques described in the − ⌘ ⌘ previous lectures. In particular, 2 1 6 mod 11 and we have x 6 mod 11 or x 30 3 mod 11. − ⌘ ⌘ ⌘ ⌘ ⌅ So let’s start by investigating the simplest such congruence: x2 a mod m. ⌘ Definition 2 Let p a prime number. Then the integer a is said to be a quadratic residue of p if the congruence x2 a mod p ⌘ has a solution. More generally, if m is any positive integer, we say that a is a quadratic residue of m if gcd(a, m)=1 and the congruence x2 a mod m has a solution. If a is not a quadratic residue it’s said to be a quadratic non-residue.
    [Show full text]
  • Modern Computer Arithmetic (Version 0.5. 1)
    Modern Computer Arithmetic Richard P. Brent and Paul Zimmermann Version 0.5.1 arXiv:1004.4710v1 [cs.DS] 27 Apr 2010 Copyright c 2003-2010 Richard P. Brent and Paul Zimmermann This electronic version is distributed under the terms and conditions of the Creative Commons license “Attribution-Noncommercial-No Derivative Works 3.0”. You are free to copy, distribute and transmit this book under the following conditions: Attribution. You must attribute the work in the manner specified • by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). Noncommercial. You may not use this work for commercial purposes. • No Derivative Works. You may not alter, transform, or build upon • this work. For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to the web page below. Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this license impairs or restricts the author’s moral rights. For more information about the license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ Contents Contents iii Preface ix Acknowledgements xi Notation xiii 1 Integer Arithmetic 1 1.1 RepresentationandNotations . 1 1.2 AdditionandSubtraction . .. 2 1.3 Multiplication . 3 1.3.1 Naive Multiplication . 4 1.3.2 Karatsuba’s Algorithm . 5 1.3.3 Toom-Cook Multiplication . 7 1.3.4 UseoftheFastFourierTransform(FFT) . 8 1.3.5 Unbalanced Multiplication . 9 1.3.6 Squaring.......................... 12 1.3.7 Multiplication by a Constant .
    [Show full text]
  • Arxiv:1408.0235V7 [Math.NT] 21 Oct 2016 Quadratic Residues and Non
    Quadratic Residues and Non-Residues: Selected Topics Steve Wright Department of Mathematics and Statistics Oakland University Rochester, Michigan 48309 U.S.A. e-mail: [email protected] arXiv:1408.0235v7 [math.NT] 21 Oct 2016 For Linda i Contents Preface vii Chapter 1. Introduction: Solving the General Quadratic Congruence Modulo a Prime 1 1. Linear and Quadratic Congruences 1 2. The Disquisitiones Arithmeticae 4 3. Notation, Terminology, and Some Useful Elementary Number Theory 6 Chapter 2. Basic Facts 9 1. The Legendre Symbol, Euler’s Criterion, and other Important Things 9 2. The Basic Problem and the Fundamental Problem for a Prime 13 3. Gauss’ Lemma and the Fundamental Problem for the Prime 2 15 Chapter 3. Gauss’ Theorema Aureum:theLawofQuadraticReciprocity 19 1. What is a reciprocity law? 20 2. The Law of Quadratic Reciprocity 23 3. Some History 26 4. Proofs of the Law of Quadratic Reciprocity 30 5. A Proof of Quadratic Reciprocity via Gauss’ Lemma 31 6. Another Proof of Quadratic Reciprocity via Gauss’ Lemma 35 7. A Proof of Quadratic Reciprocity via Gauss Sums: Introduction 36 8. Algebraic Number Theory 37 9. Proof of Quadratic Reciprocity via Gauss Sums: Conclusion 44 10. A Proof of Quadratic Reciprocity via Ideal Theory: Introduction 50 11. The Structure of Ideals in a Quadratic Number Field 50 12. Proof of Quadratic Reciprocity via Ideal Theory: Conclusion 57 13. A Proof of Quadratic Reciprocity via Galois Theory 65 Chapter 4. Four Interesting Applications of Quadratic Reciprocity 71 1. Solution of the Fundamental Problem for Odd Primes 72 2. Solution of the Basic Problem 75 3.
    [Show full text]