Algorithmic Number Theory S. Arun-Kumar December 1, 2002 2 Contents I Lectures 9 1 Lecture-wise break up 11 2 Divisibility and the Euclidean Algorithm 13 3 Fibonacci Numbers 15 4 Continued Fractions 19 5 Simple Infinite Continued Fraction 23 6 Rational Approximation of Irrationals 29 7 Quadratic Irrational(Periodic Continued Fraction) 33 8 Primes and ther Infinitude 37 9 Tchebychev’s Theorem 45 9.1 Primes and their Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 10 Linear congruences, Chinese Remainder Theorem and Fermat’s Little Theorem 51 10.1 Linear Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 10.2 Linear congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 10.3 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 10.4 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 11 Euler’s φ function, Generalisation of FLT, CRT 57 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 11.2 EULER s PHI-FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 4 CONTENTS 11.3 FERMAT’s THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 11.4 EULER s GENERALIZATION of FERMAT s THEOREM . . . . . . . . . . . . . . . . . . . . . 59 11.5 GAUSS s THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 11.6 Different Proof of CRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 11.7 Significance of CRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 12 Congrunces of Higher Degree 63 13 Lagrange’s Theorem 67 13.1 Lecture 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 13.1.1 Theorem 12.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 13.1.2 Theorem 12.2 - Lagrange’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 13.1.3 Theorem 12.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 14 Primitive Roots and Euler’s Criterion 69 14.1 Euler’s Criterion and Strengthened Euler’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 69 14.2 The Order of an Integer Modulo n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 14.3 Primitive Roots of Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 15 Quadratic Reciprocity 75 15.1 Legendre Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 15.2 Gauss’ Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 15.3 Gauss’ Reciprocity Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 16 Applications of Quadratic Reciprocity 79 17 The Jacobi Symbol 83 18 Elementary Algebraic Concepts 87 19 Sylow’s Theorem 93 20 Finite Abelian Groups & Dirichlet Characters 97 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 20.2 Characters of Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 20.3 Characters of a Finite Abelian Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 CONTENTS 5 20.4 Dirichlet Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 21 Dirichlet Products 105 22 Primes are in P 111 II Examples 115 23 Akshat Verma 117 23.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 23.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 23.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 23.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 23.5 Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 24 Rahul Gupta 121 24.1 Linear Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 24.2 Euler Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 24.3 Primitive Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 24.4 Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 24.5 Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 25 Gaurav Gupta 125 25.1 Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 25.2 Fermat’s Little theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 25.3 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 25.4 Euler’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 25.5 GCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 26 Ashish Rastogi 129 26.1 Greatest Common Divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 26.2 General Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 26.3 Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 26.4 Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 26.5 Multiplicative Functions and Perfect Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6 CONTENTS 27 Dhan Mahesh 137 27.1 Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 27.2 Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 27.3 Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 27.4 Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 27.5 Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 28 Mayank Kumar 141 28.1 GCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 28.2 Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 28.3 Euler’s Phi Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 28.4 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 28.5 Jacobi Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 29 Hitesh Chaudhary 145 29.1 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 29.2 Tchebychev’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 29.3 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 29.4 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 29.5 Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 30 Satish Parvataneni 147 30.1 CRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 30.2 FLT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 30.3 GCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 30.4 Linear Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 30.5 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 31 Bipin Tripathi 151 31.1 Euler φ function, FLT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 31.2 Congruences of higher degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 31.3 Quadratic Irrational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 31.4 Congruence, Euclidian Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 31.5 Primitive Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 CONTENTS 7 32 Amit Agarwal 155 32.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 32.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 32.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 32.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 32.5 Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 33 Vipul Jain 159 33.1 Primes and their Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 33.2 Linear Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 33.3 The Fibonacci Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 33.4 Euler’s Phi function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 33.5 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 34 Tushar Chaudhary 163 34.1 Fibonacci numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 34.2 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 34.3 Wilson’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 34.4 GCD, Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 34.5 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 35 Keshav Kunal 167 35.1 Infinitude of Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 35.2 Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 35.3 Approximation of Irrationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 35.4 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 35.5 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 36 Akrosh Gandhi 173 36.1 Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 36.2 Linear Conrguence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 36.3 Periodic Continued Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 36.4 Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 36.5 MultiplicativeFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 8 CONTENTS 37 Sai Pramod Kumar 177 37.1 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 37.2 Infinite Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 37.3 Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 37.4 Primitive Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 37.5 Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 38 Tariq Aftab 183 38.1 Congruences of higher degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 38.2 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 38.3 Euler’s Totient Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 38.4 Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 38.5 Tchebychev’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 39 Vikas Bansal 189 39.1 Generalisation of Euler’s Thoerem * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 39.2 Primes and Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 39.3 Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 39.4 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 39.5 Algebraic Number Theory (Fields) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 39.6 Greatest Integer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 40 Anuj Saxena 193 40.1 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 40.2 Euler’s φ -Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 40.3 General Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 40.4 Quadratic Residue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 40.5 Sylow Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Part I Lectures 9 [...]... 0 n→∞ hence , every irrational number uniquely represents an infinite simple continued fraction.(uniqueness follows 2 from Theorem 5.4) Corollary 5.11 For any irrational number x , |x− where Cn = pn qn 1 1 pn |< < 2 qn qn qn+1 qn is nth convergent Example 5.2 Prove that e is an irrational number Sol : Proof by contradiction, Assume that e = a , a > b > 0 is an rational number. Then for n > b and also... since they are both solutions of the equation x2 = x + 1 2 Theorem 3.7 Every positive integer may be expressed as the sun of distinct fibonacci numbers Proof: We actually prove the following claim Claim Every number in the set {1, 2, , Fn − 1} is a sum of distinct numbers from {F1 , F2 , , Fn−2 } We prove this claim by induction on n For n = 1 it is trivial Assume the claim is true for n = k Choose... Fk+1 We have N − Fk−1 < Fk+1 − Fk−1 = Fk By the induction hypothesis, N − Fk−1 is representable as a sum of distinct numbers from {F1 , F2 , , Fk−2 } By adding Fk we get that N 2 is representable as a sum of distinct numbers from {F1 , F2 , , Fk−2 , Fk−1 } 18 CHAPTER 3 FIBONACCI NUMBERS Chapter 4 Continued Fractions Definition 4.1 A continued fraction is of the form b1 a0 + b2 a1 + a2 + b3 where... finite continued fractions” Fact 4.1 Any SFCF represents a rational number Theorem 4.2 Every rational number may be expressed as a simple finite continued fraction Corollary 4.3 If 0 < a/b < 1 then a0 = 0 Fact 4.4 If a/b = [a0 ; a1 , a2 , , an ], then if an > 1, we may also write a/b = [a0 ; a1 , a2 , , an − 1, 1] Hence every rational number has at most two representations as a SFCF Example 4.2 Fn+1... expantion and the number of elements in the block is called length of the block Theorem 5.3 Every SICF represents an irrational number Proof: Let C = [a0 ; a1 , a2 , ] be a SICF and {Cn } be a sequence of convergent Clearly , for any successive convergents Cn and Cn+1 , C lies in between Cn and Cn+1 ⇒ 0 < | C − Cn | < | Cn+1 − Cn | = let us assume limit of convergent is a rational number , say ⇒ ⇒... gcd(Fm , Fn ) gcd(Fnq−1 Fr + Fnq Fr+1 , Fn ) gcd(Fnq−1 Fr , Fn ) gcd(Fr , Fn gcd(Fn , Fr since gcd(Fnq−1 , Fn ) = 1 2 Theorem 3.4 The GCD of two fibonacci numbers is again a fibonacci number Fgcd(n,m) 15 In fact, gcd(Fn , Fm ) = 16 CHAPTER 3 FIBONACCI NUMBERS Proof: Lemma 3.1 essentially tells us that something very similar to the Euclidean algorithm works here too The correpondence is made clear by the... each other Chapter 6 Rational Approximation of Irrationals In this chapter we consider the problem of finding good rational approximations to an irrational number x Definition 6.1 The best approximation to a real number x relative to n is the rational number p/q closest to x such that 0 < b ≤ n The next theorem shows that continued fraction convergents are the best approximations relative to their denominators... the SICF can be defined as the limit of the sequence of rational numbers Cn = [a0 ; a1 , a2 , , an ] (n ≥ 0 ) i.e the SICF [a0 ; a1 , a2 , ] has the value limn→∞ Cn Note : The existence of the limit in the above definition is direct from the T heorem 5.1 , T heorem 5.2 and from the fact that the subsequences of {Cn } , even and odd numbered convergents ,converge to same limit α and so {Cn } will... Fibonacci Numbers Finite Continued Fractions Simple Infinite Continued Fractions Approximations of Irrationals (Hurwitz’s theorem) Quadratic Irrationals (Periodic Continued Fractions) Primes and the Infinitude of primes π(x) Tchebychev’s theorem ( x is bounded) lnx Linear Congruences, Fermat’s little theorem and CRT Euler’s φ function, Generalization of FLT and CRT Using CRT to compute with large numbers... 0 ≤ i ≤ k 2 Corollary 5.5 Distinct continued fractions represent distinct irrationals Note : T heorem 5.3 and T heorem 5.4 together say that every SICF represents a unique irrational number Theorem 5.6 Any irrational number x can be written as [a0 ; a1 , a2 , , an−1 , xn ], where a0 is a integer ,∀i ai ∈ N and for all n xn is irrational Proof outline: By induction on n 2 Theorem 5.7 If x = [a0 ; . Algorithmic Number Theory S. Arun-Kumar December 1, 2002 2 Contents I Lectures 9 1 Lecture-wise break up 11 2 Divisibility and the Euclidean Algorithm 13 3 Fibonacci Numbers 15 4. . . . . . . . . . . . 129 26.2 General Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 26.3 Fibonacci Numbers . . . . . . . . . . . . . . gcd( F n , F r ✷ Theorem 3.4 The GCD of two fibonacci numbers is again a fibonacci number. In fact, gcd(F n , F m ) = F gcd ( n,m) . 15 16 CHAPTER 3. FIBONACCI NUMBERS Proof: Lemma 3.1 essentially tells