MAT 6932 - Number Theory Seminar Jay Pantone University of Florida Last Edited: August 19, 2012 2 Contents 1 Some Basic Theorems 1 1.1 RootsofIntegers .................................. ........ 1 1.2 SomeTheoremsofCantor ............................. ........ 3 2 Decimal Representations 4 2.1 DecimalRepresentations............................. ......... 4 2.2 Decimal Expansions and a Criterion for Irrationality . ............ 6 2.3 DecimalApproximations............................... ....... 8 2.4 Irrationality of π .......................................... 11 2.5 Irrationality of π2 .......................................... 12 2.6 Irrationality of the Trig Functions . .......... 14 2.7 Irrationality of the Hyperbolic Trig Functions . ........... 16 3 Engel Series Expansions 17 3.1 Construction ...................................... ...... 17 3.2 SomeTheorems..................................... ...... 19 4 Cantor Product Representation 22 4.1 Construction ...................................... ...... 22 4.2 SomeTheorems..................................... ...... 24 4.3 EvaluationofaSpecialCantorProduct . .......... 26 4.4 ConnectionWithEngelSeries. .. .. .. .. .. .. .. .. .. .. .. ........ 27 5 Sylvester Series 28 5.1 Construction ...................................... ...... 28 5.2 SomeTheorems..................................... ...... 30 5.3 An Irrationality Criterion . ....... 31 6 Continued Fractions 34 6.1 GeneralContinuedFractions. .......... 34 6.2 ConvergenceofGeneralContinuedFractions. .............. 36 6.3 Continued Fractions for e and π .................................. 38 6.4 ConversionofContinuedFractions . .......... 40 6.5 Bessel Functions: Their Quotients and Their Irrationals . .............. 43 6.6 SomeTheorems..................................... ...... 48 6.7 Simple Continued Fractions . ....... 51 6.8 Finite Continued Fraction Expansion of Rationals . ........... 55 6.9 Minkowiski Question Mark Function . ........ 56 6.10 Continued Fraction for e ...................................... 57 6.11 Irrational Numbers and Continued Fraction Convergence . ................. 59 7 Best Approximations 62 7.1 DefinitionandSomeTheorems ........................... ....... 62 7.2 AProceduretoGenerateBestApproximations . ............ 67 i ii CONTENTS 7.3 Connection with Continued Fractions . .......... 69 8 Equivalence of Real Numbers 70 8.1 Definition .......................................... .... 70 8.2 Connection with Continued Fractions . .......... 71 8.3 TheMarkovConstantofaRealNumber . .......... 74 9 Farey Fractions 80 10 Transcendental Numbers 85 10.1 Khintchin’s Metric Theorems . ........ 85 10.2 TheFirstTranscendentalNumber . ........... 87 11 Irrationality Type and Measure 88 11.1 Definitions........................................ ...... 88 11.2 TheThue-Siegel-Roth-DysonTheorem . ............ 90 11.3 Methods for Obtaining Irrationality Type and Measure . .............. 91 11.4 ζ(2) ................................................ 108 11.5 ζ(3) ................................................ 112 11.6 Transcendence of e ......................................... 115 11.7 Transcendence of π ......................................... 117 11.8 ζ(2π)................................................. 119 12 The Transcendence Theorems of Lindemann and Weierstrass 123 13 The Gelfond-Schneider Theorem 128 14 Transcendence Degree and Transcendental Functions 133 14.1 Definitions........................................ ...... 133 14.2 Schanuel’s Conjecture and its Implications . ............ 134 14.3 Transcendental Functions and their Exceptional Sets . ................. 135 15 Uniform Distribution 138 15.1 BasicTheorems.................................... ....... 138 15.2 An Application of Kronecker’s Theorem to Geometry . ............. 142 15.3 Simultaneous Approximation of Real Numbers . ........... 144 15.4 Kronecker’s Theorem in Two Dimensions . ......... 148 15.5 Uniform Distribution Modulo 1 . ....... 151 15.6 TheWeylCriterion.................................. ....... 154 15.7 Similar results on Q ........................................ 156 15.8 Uniform Distribution of Sequences Using Weyl’s Criterion . ............ 163 15.9 ATheoremofvanderCorput. .. .. .. .. .. .. .. .. .. .. ......... 168 15.10 SuccessiveDifferences . .. .. .. .. .. .. .. .. .. .. .. .. .......... 172 15.11 Metric Theorems on Uniform Distribution . ........... 173 16 Diophantine Approximations and Transcendence 175 16.1 Koksma’sGeneralMetricTheorem . .......... 175 16.2 ThePisot-VijayaraghavanNumbers. ............ 181 16.3 NormalNumbers ................................... ....... 183 16.4 Uniform Distribution of Integer Sequences . ............. 186 16.5 Connection Between Uniform Distribution Mod 1 of Reals and Uniform Distribution Mod m of Integer Sequences 16.6 AnApplication ...................................... ..... 195 A Frank Patane: Irrationality Measure 196 B Todd Molnar: Periodic Continued Fractions 198 CONTENTS iii C Duc Huynh: Ford’s Theorem 200 D Jay Pantone: History of π 203 E Meng Liu: Certain Trigonometric Values 220 ∞ n F Ying Guo: Transcendence of α2 223 n=0 G Ali Uncu: Ramanujan Sums 226 H Hongyan Hou: Bernoulli Coefficients 230 I Frank Patane: Bernoulli Polynomials 233 J Todd Molnar: The Sathe/Selberg-Delange Theorem 237 Index 242 iv CONTENTS Chapter 1 Some Basic Theorems 1.1 Roots of Integers Theorem 1: (Euclid) √2 is irrational. p Proof: Suppose toward a contradiction that √2 is rational. Then, √2 = for some p, q Z with q ∈ gcd(p, q) = 1. Now, p p2 √2= = 2= q ⇒ q2 = 2q2 = p2 ⇒ = p2 is even ⇒ = p is even ⇒ = p2 =4r2 =2q2, for some r Z ⇒ ∈ = 2r2 = q2 ⇒ = q2 is even ⇒ = q is even ⇒ = gcd(p, q) =1. ⇒ We have reached a contradiction. Thus √2 is not rational. Theorem 2: If n Z and n is not a perfect kth power for some k Z+, then √k n is irrational. ∈ ∈ a Proof: Suppose that √k n = , for some a,b Z. Define ν (m) := the largest power of a prime p that b ∈ p k k k k divides m. So, nb = a . Now now that for all primes p, we have νp(a ) k. Similarly, νp(b ) k. | | th Hence νp(n) k. Now, since all primes p divide n a multiple of k times, we have that n is a perfect k power. Thus| we have proved the contrapositive. Corollary: If n, k Z+, then either √k n Z or √k n is irrational. ∈ ∈ 1 2 CHAPTER 1. SOME BASIC THEOREMS Theorem 3: (Rational Root Theorem) If P (x) Z[x] is written as P (x) = a + a x + + a xn, and if ∈ 0 1 ··· n P p = 0, for p Q with gcd(p, q) = 1, then q a and p a . q q ∈ | n | 0 p p pn Proof: Let P ( )= a + a + + a n . Then, q 0 1 q ··· n q n n 1 n a q = a q − p + + a p . (3.1) − 0 1 ··· n Now observe that the right-hand side of (3.1) is a multiple of p, and thus so is the left-hand side. Since n gcd(p, q) = 1, we have p a0. Similarly, since q LHS(3.1) and q divides all terms but anp in the right-hand side of (3.1), we| must have q a pn. Since| gcd(p, q) = 1, it follows that q a . | n | n Corollary: If P (x) Z[z] is monic, then the rational roots, if any, are integers which divide a = P (0). ∈ 0 Example: P (x)= x2 2. By the Rational Root Theorem, there are no integer roots (we only need to test 1, 2). So, the two− roots must be irrational, and thus √2 is irrational. ± ± Exercise: Show that √2+ √3 is irrational. Archimedean Property: If α, β are positive reals, then there exists n N+ such that nα<β. ∈ Theorem 4: Between any two real numbers, there exists both a rational and an irrational number. Proof: Without loss of generality, let 0 <α<β. Then, β α> 0. So, there exists n Z+ such that n(β α) > 1. Now, nβ > nα + 1. Since nβ and nα differ by− more than 1, there is an∈ integer m such − m that nα<m<nβ. Thus, α< n <β. To find an irrational number between α and β, find n such that n(β α) > √2 and proceed similarly. − Corollary: Between any two rationals there exists an irrational. Between any two irrationals there exists a rational. 1.2. SOME THEOREMS OF CANTOR 3 1.2 Some Theorems of Cantor Theorem A: (Cantor) The set of rationals Q is countable. Proof: First list the rationals in [0, 1] is lexicographical order by denominator then numerator: 0 1 1 1 2 1 3 , , , , , , ,.... 1 1 2 3 3 4 4 To get all of Q, take each fraction in the list, and follow it by its negative, its reciprocal, and its negative reciprocal: 0 1 1 1 1 2 2 1 1 3 3 , , , , , , , , , , ,.... 1 1 −1 2 −2 1 −1 3 −3 1 −1 This is all of Q, enumerated on the integers. Thus Q is countable. Theorem B: (Cantor) R is uncountable. Theorem C: (Cantor) The set RQ of irrationals is uncountable. Remark: Just because two things have the same cardinality doesn’t mean it’s easy to find a bijection between them. For example, consider the sets [0, 1] and (0, 1). To find a bijection between these two sets, pick a countably infinite set X := 0, 1, x , x ,... [0, 1]. Then, let { 1 2 }⊂ r, r X x , r =0∈ ϕ(r) := 1 . x , r =1 2 xi+2, r = xi Chapter 2 Decimal Representations 2.1 Decimal Representations Definition: A decimal representation is a series ∞ a i , 10i i=1 where ai satisfies 0 ai 9 and for infinitely many ai, we have ai < 9 (this prevents duplicate series such as .6729999 ... = .673).≤ ≤ Remark: Note that: ∞ a ∞ 9 0 i < =1. ≤ 10i 10i i=1 i=1 Theorem 5: (i) Every decimal representation converges to a real number α [0, 1). ∈ (ii) Every real α [0, 1) admits a decimal representation. ∈ (iii) The decimal representation in (ii) is unique. Proof: We’ve already shown part (i). Next, denote [x] := the greatest integer x and x := x [x] [0, 1). ≤ { } − ∈ Uniqueness: Let α = .a a a .... So, 10α = a + .a a ... and thus a = [10α] and α := .a a ... = 10α . 1 2 3 1 2 3 1 1 2 3 { } So, a1 and α1 are uniquely determined. Iterating, we see that all ai are uniquely defined by a = [10α ] i+1 i . (5.1) α = 10α i+1 { i} Existence Given an α, define a as in (5.1) with α =: α . Clearly 0 a 9, since 0 α < 1. i 0 ≤ i ≤ ≤ i So 0 10αi < 10. Now we claim α = .a1a2 ..., and we use induction. Note that ≤ a 10α α α .a = α 1 = − 1 . − 1 − 10 10 4 2.1. DECIMAL REPRESENTATIONS 5 10an 1 Now we claim that α .a a ...a = { − }. To see this, observe that − 1 2 n 10n a1 an 1 an − α .a1a2 ...an = α + + n 1 n . − − 10 ··· 10 − − 10 By the Induction Hypothesis, this equals 10αi 2 an 10αn 1 [10αn 1] 10αn 1 − − − − { n 1 } n = − n = { n }. 10 − − 10 10 10 Now, by letting n , we get existence.