Pell and Pell–Lucas Numbers with Applications

Thomas Koshy

Pell and Pell–Lucas Numbers with Applications

123 Thomas Koshy Framingham State University Framingham, MA, USA

ISBN 978-1-4614-8488-2 ISBN 978-1-4614-8489-9 (eBook) DOI 10.1007/978-1-4614-8489-9 Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2013956605

Mathematics Subject Classification (2010): 03-XX, 11-XX

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Cover graphic: Pellnomial binary triangle

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com) Dedicated to A good friend, colleague, an excellent problem-proposer and solver, and a Fibonacci and Pell enthusiast, Thomas Eugene Moore (1944–2014) of Bridgewater State University, Bridgewater, Massachusetts

Preface

A thing of beauty is a joy for ever: Its loveliness increases; It will never pass into nothingness. – John Keats (1777–1855)

Two New Bright Stars

Like the well-known Fibonacci and Lucas numbers, Pell and Pell–Lucas numbers are two spectacularly bright stars on the mathematical firmament. They too continue to amaze the mathematical community with their splendid beauty, ubiquity, and applicability, providing delightful opportunities to experiment, explore, conjecture, and problem-solve. Pell and Pell– Lucas numbers form a unifying thread intertwining analysis, geometry, trigonometry, and various areas of discrete mathematics, such as number theory, graph theory, linear algebra, and combinatorics. They belong to an extended Fibonacci family, and are a powerful tool for extracting numerous interesting properties of a vast array of number sequences. Both families share numerous fascinating properties.

A First in the Field

Pell and Pell–Lucas numbers and their delightful applications appear widely in the literature, but unfortunately they are scattered throughout a multitude of periodicals. As a result, they remain out of reach of many mathematicians and amateurs. This vacuum inspired me to create this book, the first attempt to collect, organize, and present information about these integer families in a systematic and enjoyable fashion. It is my hope that this unique undertaking will offer a thorough introduction to one of the most delightful topics in discrete mathematics.

Audience

The book is intended for undergraduate/graduate students depending on the college or university and the instructors in those institutions. It will also engage the intellectually curious high schoolers and teachers at all levels. The exposition proceeds from the basics to more advanced topics, motivating with examples and exercises in a rigorous, systematic fashion. Like the Catalan and Fibonacci books, this will be an important resource for seminars, independent study, and workshops.

vii viii Preface

The professional mathematician and computer scientist will certainly profit from the exposure to a variety of mathematical skills, such as pattern recognition, conjecturing, and problem-solving techniques. Through my Fibonacci and Catalan books, I continue to hear from a number of enthusiasts coming from a wide variety of backgrounds and interests, who express their rewarding experi- ences with these books. I now encourage all Pell and Pell–Lucas readers to also communicate with me about their experiences with the Pell family.

Prerequisites

This book requires a strong foundation in precalculus mathematics; users will also need a good background in matrices, determinants, congruences, combinatorics, and calculus to enjoy most of the material. It is my hope that the material included here will challenge both the mathematically sophisticated and the less advanced. I have included fundamental topics such as the floor and ceiling functions, summation and product notations, congruences, recursion, pattern recognition, generating functions, binomial coefficients, Pascal’s triangle, binomial theorem, and Fibonacci and Lucas numbers. They are briefly summarized in Chapter 1. For an extensive discussion of these topics, refer to my Elementary Number Theory with Applications and Discrete Mathematics with Applications.

Historical Background

The personalities and history behind the mathematics make up an important part of this book. The study of Pell’s equation, continued fractions, and square-triangular numbers lead into the study of the Pell family in a logical and natural fashion. The book also contains an intriguing array of applications to combinatorics, graph theory, geometry, and mathematical puzzles. It is important to note that Pell and Pell–Lucas numbers serve as a bridge linking number theory, combinatorics, graph theory, geometry, trigonometry, and analysis. These numbers occur, for example, in the study of lattice walks, and the tilings of linear and circular boards using unit square tiles and dominoes.

Pascal’s Triangle and the Pell Family

It is well known that Fibonacci and Lucas numbers can be read directly from Pascal’s triangle. Likewise, we can extract Pell and Pell–Lucas numbers also from Pascal’s triangle, showing the close relationship between the triangular array and the Pell family. Preface ix

ANewHybridFamily

The closely-related Pell and Fibonacci families are employed to construct a new hybrid Pell– Fibonacci family. That too is presented with historical background.

Opportunities for Exploration

Pell and Pell–Lucas numbers, like their closely related cousins, offer wonderful opportunities for high-school, undergraduate and graduate students to enjoy the beauty and power of mathematics, especially number theory. These families can extend a student’s mathematical horizons, and offer new, intriguing, and challenging problems. To faculty and researchers, they offer the chance to explore new applications and properties, and to advance the frontiers of mathematical knowledge. Most of the chapters end in a carefully prepared set of exercises. They provide opportunities for establishing number-theoretic properties and enhancement of problem-solving skills. Starred exercises indicate a certain degree of difficulty. Answers to all exercises can be obtained electronically from the publisher.

Symbols and Abbreviations

For quick reference, a list of symbols and a glossary of abbreviations is included. The symbols index lists symbols used, and their meanings. Likewise, the abbreviations list provides a gloss for the abbreviations used for brevity, and their meanings.

Salient Features

The salient features of the book include extensive and in-depth coverage; user-friendly approach; informal and non-intimidating style; plethora of interesting applications and properties; his- torical context, including the name and affiliation of every discoverer, and year of discovery; harmonious linkage with Pascal’s triangle, Fibonacci and Lucas numbers, Pell’s equation, continued fractions, square-triangular, pentagonal, and hexagonal numbers; trigonometry and complex numbers; Chebyshev polynomials and tilings; and the introduction of the brand-new Pell–Fibonacci hybrid family.

Acknowledgments

In undertaking this extensive project, I have immensely benefited from over 250 sources, a list of which can be found in the References. Although the information compiled here does not, of course, exhaust all applications and occurrences of the Pell family, these sources provide, to the best of my ability, a reasonable sampling of important contributions to the field. x Preface

I have immensely benefited from the constructive suggestions, comments, support, and cooperation from a number of well-wishers. To begin with, I am greatly indebted to the reviewers for their great enthusiasm and suggestions for improving drafts of the original version. I am also grateful to Steven M. Bairos of Data Translation, Inc. for his valuable comments on some early chapters of the book; to Margarite Landry for her superb editorial assistance and patience; to Jeff Gao for creating the Pascal’s binary triangle in Figure 5.6, preparing the Pell, Pell–Lucas, Fibonacci, and Lucas tables in the Appendix, and for co-authoring with me several articles on the topic; to Ann Kostant, Consultant and Senior Advisor at Springer for her boundless enthusiasm and support for the project; and to Elizabeth Loew, Senior Editor at Springer along with her Springer staff for their dedication, cooperation, and interaction with production to publish the book in a timely fashion.

Framingham, Massachusetts Thomas Koshy August, 2014 [email protected]

If I have been able to see farther, it was only because I stood on the shoulders of giants. – Sir Isaac Newton (1643–1727) Contents

Preface...... vii

ListofSymbols...... xxi

Abbreviations ...... xxiii

1 Fundamentals ...... 1

1.1 Introduction...... 1 1.2 Floor and Ceiling Functions ...... 1 1.3 SummationNotation ...... 2 1.4 Product Notation ...... 3 1.5 Congruences ...... 3 1.6 Recursion ...... 4 1.7 SolvingRecurrences ...... 8 1.7.1 LHRWCCs...... 9 1.8 GeneratingFunctions ...... 11 1.9 BinomialCoefficients...... 14 1.9.1 Pascal’s Identity ...... 14 1.9.2 BinomialTheorem...... 15 1.9.3 Pascal’sTriangle...... 17 1.10 Fibonacci and Lucas Numbers...... 18 1.10.1 Fibonacci’s Rabbits ...... 18 1.10.2 Fibonacci Numbers ...... 18 1.10.3 LucasNumbers ...... 18 1.10.4 Binet’sFormulas...... 19 1.10.5 Fibonacci and Lucas Identities ...... 19 1.10.6 Lucas’ Formula for Fn ...... 21 1.11 PellandPell–LucasNumbers:A Preview ...... 23 1.11.1 Binet-likeFormulas ...... 23 1.11.2 Example1.7Revisited...... 24 1.12 MatricesandDeterminants ...... 24 1.12.1 Matrix Addition...... 25 1.12.2 Scalar Multiplication ...... 26 1.12.3 Matrix Multiplication...... 26

xi xii Contents

1.12.4 InvertibleMatrix...... 26 1.12.5 Determinants...... 27 1.12.6 Laplace’s Expansion...... 27 Exercises1 ...... 28 2 Pell’sEquation...... 31

2.1 Introduction...... 31 2.2 Pell’s Equation x2  dy2 D .1/n ...... 38 2.3 Normofa QuadraticSurd ...... 38 2.4 RecursiveSolutions ...... 39 2.4.1 A Second-Order Recurrence for .xn;yn/ ...... 40 2.5 Solutions of x2  2y2 D .1/n ...... 40 2.5.1 An Interesting Byproduct ...... 41 2.6 Euler and Pell’s Equation x2  dy2 D .1/n ...... 41 2.7 A Link Between Any Two Solutions of x2  dy2 D .1/n ...... 48 2.8 A Preview of Chebyshev Polynomials ...... 52 2.9 Pell’s Equation x2  dy2 D k ...... 52 Exercises2 ...... 54 3 ContinuedFractions...... 57

3.1 Introduction...... 57 3.2 FiniteContinuedFractions...... 58 3.2.1 Convergentsofa ContinuedFraction...... 61

3.2.2 Recursive Definitions of pk and qk ...... 62 3.3 LDEsandContinuedFractions...... 63 3.4 InfiniteSimpleContinuedFractions(ISCF) ...... 64 3.5 Pell’s Equation x2  dy 2 D .1/n andISCFs ...... 68 3.6 A Simple Continued Fraction Tiling Model ...... 74 3.6.1 A Fibonacci Tiling Model ...... 76 3.6.2 A Pell Tiling Model ...... 76 3.7 A Generalized Continued Fraction Tiling Model ...... 77 Exercises3 ...... 77 4 PythagoreanTriples ...... 79

4.1 Introduction...... 79 4.2 PythagoreanTriples...... 80 4.2.1 PrimitivePythagoreanTriples ...... 80 4.2.2 SomeQuickObservations...... 82 4.3 A RecursiveAlgorithm ...... 83 Exercises4 ...... 84 Contents xiii

5 Triangular Numbers ...... 87

5.1 Introduction...... 87 5.2 Triangular Numbers ...... 87 5.3 Pascal’sTriangleRevisited ...... 89 5.4 Triangular Mersenne Numbers ...... 90 5.5 Properties of Triangular Numbers ...... 91 5.6 Triangular Fermat Numbers ...... 94 5.7 The Equation x2 C .x C 1/2 D z2 Revisited ...... 94 5.8 A Generating function For Triangular Numbers ...... 95 5.9 Triangular Numbers and Pell’s Equation...... 95 5.9.1 TwoInterestingDividends ...... 96 5.9.2 The Matrix Method Using Mathematicar ...... 98 5.9.3 Example5.3Revisited...... 99 5.10 AnUnsolvedProblem...... 100 Exercises5 ...... 100 6 Square-Triangular Numbers...... 101

6.1 Introduction...... 101 6.2 Infinitude of Square-Triangular Numbers ...... 101 6.2.1 AnAlternateMethod...... 105 2 6.2.2 The Ends of xk; yk; yk,andnk ...... 106 6.2.3 Cross’ Recurrence for yk ...... 107 6.3 The Infinitude of Square-Triangular Numbers Revisited ...... 110 6.4 A Recursive Definition of Square-Triangular Numbers ...... 110 6.5 Warten’s Characterization of Square-Triangular Numbers ...... 111 6.6 A Generating Function For Square-Triangular Numbers...... 113 6.6.1 A Generating Function For fnkg ...... 114 6.6.2 A Generating Function For fykg ...... 114 Exercises6 ...... 114 7 PellandPell–LucasNumbers...... 115

7.1 Introduction...... 115 7.2 EarlierOccurrences ...... 115 7.3 RecursiveDefinitions ...... 116 7.4 Alternate Forms for  and ı ...... 117 7.5 A GeometricConfluence...... 118 7.6 Pell’s equation x2  2y2 D1 Revisited...... 119 7.7 Fundamental Pell Identities ...... 122 7.7.1 Two Interesting Byproducts ...... 127 xiv Contents

7.8 PellNumbersandPrimitivePythagoreanTriples...... 129 7.9 A HarmonicBridge ...... 129 7.10 Square-Triangular Numbers with Pell Generators ...... 130 7.11 PrimitivePythagoreanTriplesWithConsecutiveLegsRevisited...... 133 7.12 Squareofa PellSum...... 134

7.13 The Recurrence xnC2 D 6xnC1  xn C 2 Revisited...... 136 7.14 RatiosofConsecutivePellandPell–LucasNumbers ...... 138 7.15 Triangular Pell Numbers ...... 138 7.16 Pentagonal Numbers ...... 140 7.17 Pentagonal Pell Numbers ...... 141 7.18 Zeitlin’s Identity ...... 143 7.19 Pentagonal Pell–Lucas Numbers ...... 144 7.20 Heptagonal Pell Numbers ...... 145 Exercises7 ...... 147 8 Additional Pell Identities ...... 151

8.1 Introduction...... 151 8.2 An Interesting Byproduct ...... 152 8.3 A PellandPell–LucasHybridity ...... 156 8.4 MatricesandPellNumbers...... 157 p 8.5 Convergents of the ISCF of 2 Revisited ...... 160 8.5.1 AnAlternateMethod...... 161 8.6 Additional Addition Formulas ...... 163 8.6.1 Formula(8.10)Revisited...... 164 8.7 Pell Divisibility Properties Revisited ...... 166 8.8 Additional Identities ...... 168 8.9 Candido’s Identity and the Pell Family...... 169 8.10 PellDeterminants ...... 170 Exercises8 ...... 171 9 Pascal’sTriangleandthePellFamily...... 173

9.1 Introduction...... 173 9.2 AnAlternateApproach ...... 177

9.3 Another Explicit Formula for Qn ...... 178 9.4 A RecurrenceforEven-numberedPellNumbers...... 179

9.5 Another Explicit Formula for P2n ...... 180

9.6 An Explicit Formula for P2n1 ...... 180 2 2 9.7 Explicit Formulas for Pn and Qn ...... 181 Contents xv

9.8 Lockwood’s Identity ...... 182 9.9 LucasNumbersandPascal’sTriangle...... 185 9.10 Pell–LucasNumbersandPascal’sTriangle...... 186 9.11 Odd-Numbered Fibonacci Numbers and Pascal’s Triangle ...... 187 9.12 Odd-NumberedPellNumbersandPascal’sTriangle...... 188 9.13 PellSummationFormulas ...... 188 Exercises9 ...... 191 10 Pell Sums and Products ...... 193

10.1 Introduction...... 193 10.2 PellandPell–LucasSums ...... 193 10.3 InfinitePellandPell–LucasSums...... 195 10.4 A Pell Inequality ...... 199 10.5 An Infinite Pell Product ...... 200 10.6 RadiiofConvergenceoftheSeries...... 202 P1 n 10.6.1 Sum of the Series Pnx ...... 202 nD0 P1 n 10.6.2 Sum of the Series Qnx ...... 204 nD0 Exercises10...... 204 11 GeneratingFunctionsforthePellFamily ...... 207

11.1 Introduction...... 207 11.2 GeneratingFunctionsforthePellandPell–LucasSequences ...... 207 Pn Pn 11.3 Formulas for A2k and A2kC1 ...... 210 kD0 kD0 Pn 11.4 A Formula for AkAnk ...... 211 kD0 Pn 11.5 A Formula for A2kA2n2k ...... 212 kD0 Pn 11.6 A Formula for A2kC1A2n2kC1 ...... 214 kD0 Pn 11.7 A Formula for the Hybrid Sum A2kA2n2kC1 ...... 215 kD0 2 2 11.8 Generating Functions for fPn g and fQng ...... 216 11.9 Generating Functions for fP2nC1g; fQ2ng; fQ2nC1g,andfP2ng Revisited ...... 217 xvi Contents

11.10 Generating Functions for fPnPnC1g and fQnQnC1g ...... 218

11.11 Another Explicit Formula for Pn ...... 218 11.12 Hoggatt’s Array ...... 222 Exercises11...... 224 12 PellWalks...... 227

12.1 Introduction...... 227 12.2 Interesting Byproducts ...... 231 12.3 Walks Beginning with and Ending in E...... 235 12.4 Paths Beginning with E and Ending in W ...... 236 12.5 Paths Beginning with E, but not Ending in W ...... 237 12.6 Paths not Beginning with or Ending in E ...... 238 12.7 A HiddenTreasure ...... 240 12.8 Example12.2Revisited...... 244 13 PellTriangles ...... 247

13.1 Introduction...... 247 13.2 CentralElementsinthePellTriangle ...... 248 13.3 An Alternate Formula for g.n; j/ ...... 249

13.4 A Recurrence for Kn ...... 251 13.5 DiDomenico’sTriangles...... 252 Exercises13...... 253 14 Pell and Pell–Lucas Polynomials ...... 255

14.1 Introduction...... 255 14.2 SpecialCases...... 256 14.3 Gauthier’sFormula...... 257 14.4 Binet-likeFormulas ...... 258 14.5 A Pell Divisibility Test ...... 260

14.6 Generating Functions for pn.x/ and qn.x/ ...... 263

14.7 Elementary Properties of pn.x/ and qn.x/...... 264 14.8 SummationFormulas ...... 264

14.9 Matrix Generators for pn.x/ and qn.x/ ...... 265 14.10 Addition Formulas ...... 266

14.11 Explicit Formulas for pn.x/ and qn.x/ ...... 270 14.12 Pell Polynomials from Rising Diagonals...... 272 14.13 Pell–Lucas Polynomials from Rising Diagonals...... 272 14.14 SummationFormulas ...... 275 Contents xvii

14.15 Pell Polynomials and Pythagorean Triples ...... 277 14.16 PythagoreanTripleswithPellGenerators...... 277 Exercises14...... 279 15 Pellonometry ...... 283

15.1 Introduction...... 283 15.2 Euler’sandMachin’sFormulas...... 285 15.3 Identities (15.1) and (15.2) Revisited ...... 285 15.4 An Additional Byproduct of Example 15.2 ...... 285 15.5 Shapiro’sFormula...... 288 15.6 Seiffert’sFormulas...... 290 15.6.1 AdditionalSeiffertFormulas...... 293  15.7 Roelants’ Expansions of 4 ...... 296 15.7.1 SpecialCases ...... 298

15.8 Another Explicit Formula for Pn ...... 298

15.9 pn.x/; qn.x/; andHyperbolicFunctions...... 300 Exercises15...... 300 16 PellTilings...... 303

16.1 Introduction...... 303 16.2 A Combinatorial Model for Fibonacci Numbers ...... 303 16.3 A Fibonacci Tiling Model ...... 305 16.4 A CombinatorialModelForPellNumbers ...... 305 16.5 ColoredTilings...... 308 16.6 CombinatorialModelsforPell–LucasNumbers...... 310 16.7 ColoredTilingsRevisited...... 313 16.8 CircularTilingsandPell–LucasNumbers ...... 315

16.9 Combinatorial Models for the Pell Polynomial pn.x/...... 318 16.10 Colored Tilings and Pell Polynomials ...... 320 16.11 Combinatorial Models for Pell–Lucas Polynomials ...... 321 16.12 Bracelets and Pell–Lucas Polynomials ...... 322 17 Pell–Fibonacci Hybridities ...... 325

17.1 Introduction...... 325 17.2 A Fibonacci Upper bound...... 325 17.3 Cook’s Inequality...... 328 17.4 Pell–Fibonacci Congruences...... 331 17.4.1 A Generalization...... 334 17.5 Israel’s Congruence ...... 334 xviii Contents

17.6 Seiffert’s Congruence ...... 335 17.6.1 Israel’s and Seiffert’s Congruences Revisited ...... 336 17.7 Pell–Lucas Congruences ...... 336 17.8 Seiffert’s Pell–Lucas Congruences ...... 338 17.9 HybridSums...... 339 17.9.1 WeightedHybridSums...... 341 17.10 Congruence Byproducts...... 342 17.10.1 SpecialCases ...... 343 17.11 A Counterpart for Pell–Lucas Numbers...... 344 17.11.1 SpecialCases ...... 345 17.12 Catalani’s Identities ...... 349 17.13 A Fibonacci–Lucas–Pell Bridge ...... 351

17.14 Recurrences for fFnPng, fLnPng, fFnQng,andfLnQng ...... 352

17.15 Generating Functions for fAng; fBng; fCng; and fDng ...... 353 17.16 ISCFRevisited ...... 356 17.16.1 SpecialCases ...... 357 17.17 BasicGraph-theoreticTerminology...... 357 17.18 Cartesian Product of Two Graphs ...... 359

17.19 Domino Tilings of W4  Pn1 ...... 360 Exercises17...... 361 18 AnExtendedPellFamily...... 363

18.1 Introduction...... 363 18.2 AnExtendedPellFamily ...... 363 18.3 A GeneralizedCassini-likeFormula ...... 365 18.3.1 TwoInterestingSpecialCases ...... 368 19 Chebyshev Polynomials ...... 371

19.1 Introduction...... 371 19.2 Chebyshev Polynomials of the First Kind ...... 371 19.3 Pell–LucasNumbersRevisited...... 372

19.4 An Explicit Formula for Tn.x/ ...... 373

19.5 Another Explicit Formula for Tn.x/...... 374 19.5.1 Two Interesting Byproducts ...... 375 2 2 2 19.6 Tn.x/ and the Pell Equation u  .x  1/v D 1 ...... 375

19.7 Chebyshev Polynomials Tn.x/ and Trigonometry ...... 376 19.8 ChebyshevRecurrenceRevisited...... 377

19.9 A Summation Formula For Tn.x/ ...... 378 19.9.1 A Summation Formula For Qn ...... 380 Contents xix

19.10 Chebyshev Polynomials of the Second Kind ...... 381 19.11 PellNumbersRevisited...... 382

19.12 An Explicit Formula for Un.x/ ...... 382

19.13 Another Explicit Formula for Un.x/ ...... 383 19.13.1 An Explicit Formula for Pn ...... 384 19.14 Pell’sEquationRevisited ...... 385

19.15 Un.x/ and Trigonometry ...... 386

19.16 Chebyshev Recurrence for Un.x/ Revisited ...... 387 19.16.1 AnInterestingSpecialCase...... 387 19.16.2 An Interesting Byproduct ...... 389 19.17 Cassini-like Formulas for Chebyshev Polynomials...... 392 19.18 Generating Functions for Chebyshev Polynomials ...... 392 Exercises19...... 393 20 ChebyshevTilings...... 395

20.1 Introduction...... 395

20.2 Combinatorial Models for Un.x/ ...... 395

20.3 A Colored Combinatorial Model for Un.x/...... 399

20.4 Combinatorial Models for Tn.x/ ...... 400

20.5 A Combinatorial Proof that Tn.cos / D cos n ...... 403 20.6 Two Hybrid Chebyshev Identities ...... 405 Exercises20...... 410 Appendix...... 411

References...... 417

Index...... 423

List of Symbols

Symbol Meaning bxc greatest integer Ä x dxe least integer  x end of an example, and for statements of theorems, lemmas and corollaries without proofs. iPDm Pm ai D ai ak C akC1 CCam iDk iDk iQDm Qm ai D ai ak akC1 am iDk iDk ajbais a factor of b a Á b.mod m/ a is congruent to b a 6Á b.mod m/ a is not congruent to b nŠ n.n  1/  3  2  1 W set of whole numbers 0; 1; 2; : : : .a1;a2;:::;an/ greatest common factor of a1;a2;:::;an n Mn nth Mersenne number 2  1 .a1a2 an/two binary number a1a2 an n r binomial coefficient Fn nth Fibonacci number Ln nth Lucasp number ˛.1Cp5/=2 ˇ.1 5/=2 Pn nth Pell number Ln nth Pell–Lucasp number 1Cp2 ı1 2 jxj absolute value of real number x A D .aij /mn matrix A of size m by n In identity matrix of size n by n A1 inverse of square matrix A jAj determinant of square matrix A

lg x log2 x x2  dy2 D 1 Pell’s equation x2  dy2 D .1/n Pell’s equation

xxi xxii List of Symbols

Symbol Meaning p u D x C y d quadraticp surd ux y d N.u/ x2  dy2  is approximately equal to fang sequence with nth term an AB line segment AB x2  dy2 D k Pell’s equation Œa0I a1;a2;:::;an finite simple continued fraction Ck kth convergent Œa0I a1;a2;:::;an infinite simple continued fraction x-y-z Pythagorean triple tn nth triangular number ? unsolved problem .N/ digital root of N a . m / Jacobi symbol pn nth Pentagonal number  eigenvalue a6 j bais not a factor of b Œa; b pleast common multiple of a and b i 1 pn.x/ nth Pell polynomial in x qn.x/ nth Pell–Lucasp polynomial in x 2 ˛.x/ .x Cpx C 4/=2 2 ˇ.x/ .x p x C 4/=2 2 .x/ x Cpx C 1 ı.x/ x  x2 C 1 Tn.x/ nth Chebyshev polynomial of the first kind Un.x/ nth Chebyshev polynomial of the second kind Abbreviations

Abbreviation Meaning LHRWCCs linear homogeneous recurrence with constant coefficients LNHRWCCs linear nonhomogeneous recurrence with constant coefficients RHS right-hand side LHS left-hand side PMI principle of mathematical induction rms root-mean-square AIME American Invitational Mathematics Examinations FSCF finite simple continued fraction LDE linear diophantine equation ISCF infinite simple continued fraction

xxiii