An Introduction to Number Theory J. J. P. Veerman June 27, 2021 © 2020 J. J. P. Veerman This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) You are free to: Share—copy and redistribute the material in any medium or format Adapt—remix, transform, and build upon the material for any purpose, even commercially. The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution—You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions—You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Non-Commercial—You may not use the material for commercial purposes. List of Figures p 1 Eratosthenes’ sieve up to n = 30. All multiples of a less than 31 are cancelled. The remainder are the primes less than n = 31. 4 2 A directed path g passing through all points of Z2. 12 R x 3 On the left, the function 2 lnt dt in blue, p(x) in red, and R x x=lnx in green. On the right, we have 2 lnt dt − x=lnx in blue, p(x) − x=lnx in red. 29 ¥ R ¥ −s 4 Proof that ∑n=1 f (n) is greater than 1 x dx if f is positive and decreasing. 34 ¥ 5 Proof that ∑n=1 f (n) (shaded in blue and green) minus f (1) R ¥ −s (shaded in blue) is less than 1 x dx if f is positive and decreasing to 0. 37 6 The origin is marked by “×”. The red dots are visible from ×; between any blue dot and × there is a red dot. The picture shows exactly one quarter of {−4;··· ;4g2n(0;0) ⊂ Z2. 38 7 The general solution of the inhomogeneous equation (~r;~x) = c in R2. 45 8 A one parameter family ft of maps from the circle to itself. For every t 2 [0;1] the map ft is constructed by truncating the map x ! 2x mod 1 as indicated in this figure. 65 3 4 List of Figures 9 Three branches of the Gauss map. 90 10 The line y = wx and (in red) successive iterates of the rotation Rw . Closest returns in this figure are q in f2;3;5;8g. 99 11 The geometry of successive closest returns. 99 12 Drawing y = w1x and successive approximations (an+1 is taken to be 3). The green arrows correspond to en−1, en, and en+1. 100 13 Black: thread from origin with golden mean slope; red: pulling the thread down from the origin; green: pulling the thread up from the origin. 103 14 Plots of the points (n;n) in polar coordinates, for n ranging from 1 to 50, 180, 330, and 3000, respectively. 106 15 Plots of the prime points (p; p) (p prime) in polar coordinates with p ranging between 2 and 3000, and between 2 and 30000, respectively. 107 16 The Gaussian integers are the lattice points in the complex plane; both real and imaginary parts are integers. For an arbitrary point z 2 C — marked by x in the figure, a nearby integer is k1 + ik2 where k1 is the closest integer to Re(z) and k2 the closest integer to Im(z). In this case that is 2 + 3i. 121 p 17 A depiction of Z[ −6] in the complex plane;p real parts are integers and imaginary parts are multiples of 6. 122 18 The part to the right of the intersection with ` : y = x + 1 (dashed) of bad path (in red) is reflected. The reflected part in indicated in green. The path becomes a monotone path from (0;0) to (n − 1;n + 1). 124 19 The Gaussian primes described in Proposition 7.30. There are approximately 950 within a radius 40 of the origin (left figure) and about 3300 within a radius 80 (right figure). 127 20 Possible values of rg−1 in the proof of Proposition 7.26. 129 21 A comparison between approximating the Lebesgue integral (left) and the Riemann integral (right). 138 22 The pushforward of a measure n. 139 − + 23 The functions m(Xc ) and m(Xc ). 140 List of Figures 5 24 This map is not uniquely ergodic 142 25 The first two stages of the construction of the singular measure np. The shaded parts are taken out. 145 26 The first two stages of the construction of the middle third Cantor set. 148 27 The inverse image of a small interval dy is T −1(dy) 154 p 28 w is irrational and q is a convergent of w. Then x + qw modulo 1 is close to x. Thus adding qw modulo 1 amounts to a translation by a small distance. Note: “om” in the figure stands for w. 157 1 1 29 `(I) is between 3 and 2 of `(J). So there are two disjoint images −1 of I under Rw that fall in J. 158 30 An example of the system described in Corollary 9.9. 160 31 Illustration of the fact that for a concave function f , we have f (wx + (1 − w)y) ≥ w f (x) + (1 − w) f (y) (Jensen’s inequality). 168 32 Plot of the function ln(x)ln(1 + x) 169 33 Left, a curve. Then two simple, closed curves with opposite orientation. The curve on the right is a union of two simple, closed curves. 175 34 A singular point z0 where f is bounded does not contribute to H g f dz. 176 35 The curve g goes around z exactly once in counter-clockwise direction. If d is small enough, z + d also lies inside g. 179 36 The curve w goes around z0 exactly once in counter-clockwise direction. 181 37 g is analytic in DR := fRez ≥ −dRg \ fjzj ≤ Rg (shaded). The red is p p curve is given by C+(s) = Re with s 2 (− 2 ; 2 ). The green curve is p 3p is given by C+(s) = Re with s 2 ( 2 ; 2 ). The blue L− consists of 2 small circular segments plus the segment connecting their left endpoints at a distance d to the left of the the imaginary axis. 182 38 The complex plane with eit , −e−it and e−it on the unit circle. cost is the average of eit and e−it and isint as the average of eit and −e−it . 186 6 List of Figures 39 The functions gi and hi of exercise 10.20 for i 2 f2;8;15;30g. 190 40 The contour C is the concatenation of c (celeste), b1 (blue), r1 (red), g (green), p (purple), −g, r2, and b2. The path r is a semi-circle of radius R. The path p is a small circle of radius r. See exercise 10.21. 190 41 Integration over the shaded triangle of area 1=2 in equation 11.9. 199 42 The functions q(x)=x (green), y(x)=x (red), and p(x)lnx=x (blue) for x 2 [1;1000]. All converge to 1 as x tends to infinity. The x-axis is horizontal. 208 Contents List of Figures 3 Part 1. Introduction to Number Theory Chapter 1. A Quick Tour of Number Theory 3 x1.1. Divisors and Congruences 4 x1.2. Rational and Irrational Numbers 5 x1.3. Algebraic and Transcendental Numbers 7 x1.4. Countable and Uncountable Sets 9 x1.5. Exercises 13 Chapter 2. The Fundamental Theorem of Arithmetic 19 x2.1. Bezout’s´ Lemma 20 x2.2. Corollaries of Bezout’s´ Lemma 21 x2.3. The Fundamental Theorem of Arithmetic 23 x2.4. Corollaries of the Fundamental Theorem of Arithmetic 25 x2.5. The Riemann Hypothesis 27 x2.6. Exercises 30 Chapter 3. Linear Diophantine Equations 39 x3.1. The Euclidean Algorithm 39 7 8 Contents x3.2. A Particular Solution of ax + by = c 41 x3.3. Solution of the Homogeneous equation ax + by = 0 43 x3.4. The General Solution of ax + by = c 44 x3.5. Recursive Solution of x and y in the Diophantine Equation 46 x3.6. Exercises 47 Chapter 4. Number Theoretic Functions 55 x4.1. Multiplicative Functions 55 x4.2. Additive Functions 58 x4.3. Mobius¨ inversion 58 x4.4. Euler’s Phi or Totient Function 60 x4.5. Dirichlet and Lambert Series 62 x4.6. Exercises 65 Chapter 5. Modular Arithmetic and Primes 73 x5.1. Modular Arithmetic 73 x5.2. Euler’s Theorem 74 x5.3. Fermat’s Little Theorem and Primality Testing 76 x5.4. Fermat and Mersenne Primes 79 x5.5. Division in Zb 81 x5.6. Exercises 83 Chapter 6. Continued Fractions 89 x6.1. The Gauss Map 89 x6.2. Continued Fractions 90 x6.3. Computing with Continued Fractions 95 x6.4. The Geometric Theory of Continued Fractions 97 x6.5. Closest Returns 99 x6.6. Exercises 101 Part 2. Topics in Number Theory Chapter 7. Algebraic Integers 111 x7.1. Rings and Fields 111 Contents 9 x7.2. Primes and Integral Domains 113 x7.3. Norms 115 x7.4. Euclidean Domains 118 x7.5. Example and Counter-Example 120 x7.6. Exercises 123 Chapter 8. Ergodic Theory 133 x8.1. The Trouble with Measure Theory 133 x8.2. Measure and Integration 135 x8.3. The Birkhoff Ergodic Theorem 139 x8.4.