Pell and Pell-Lucas Numbers with Applications

Pell and Pell-Lucas numbers, like the well-known Fibonacci and Catalan numbers, continue to intrigue the mathematical world with their beauty and applicability. They offer opportunities for experimentation, exploration, conjecture, and problem-solving techniques, connecting the fields of analysis, geometry, trigonometry, and various areas of discrete mathematics, number theory, graph theory, linear algebra, and combinatorics. Pell and Pell-Lucas numbers belong to an extended Fibonacci family as a powerful tool for extracting numerous interesting properties of a vast array of number sequences.
A key feature of this work is the historical flavor that is interwoven into the extensive and in-depth coverage of the subject. An interesting array of applications to combinatorics, graph theory, geometry, and intriguing mathematical puzzles is another highlight engaging the reader. The exposition is user-friendly, yet rigorous, so that a broad audience consisting of students, math teachers and instructors, computer scientists and other professionals, along with the mathematically curious will all benefit from this book.
Finally, Pell and Pell-Lucas Numbers provides enjoyment and excitement while sharpening  the reader's mathematical skills involving pattern recognition, proof-and-problem-solving techniques.

List of contents

Preface.- List of Symbols.- Abbreviations.- 1. Fundamentals.- 2. Pell's Equation.- 3. Continued Fractions.- 4. Pythagorean Triples.- 5. Triangular Numbers.- 6. Square-Triangular Numbers.- 7. Pell and Pell-Lucas Numbers.- 8. Additional Pell Identities.- 9. Pascal's Triangle and the Pell Family.- 10. Pell Sums and Products.- 11. Generating Functions for the Pell Family.- 12. Pell Walks.- 13. Pell Triangles. - 14. Pell and Pell-Lucas Polynomials.- 15. Pellonometry.- 16. Pell Tilings.- 17. Pell-Fibonacci Bridges.- 18. An Extended Pell Family.- 19. Chebyshev Polynomials.- 20. Chebyshev Tilings.- Appendix.- References.- Index.

About the author

Thomas Koshy is Professor Emeritus of Mathematics at Framingham State University in Framingham, Massachusetts. In 2007, he received the Faculty of the Year Award and his publication Fibonacci and Lucas numbers with Applicationswon the Association of American Publishers' new book award in 2001.Professor Koshyhas also authored numerous articles on a wide spectrum of topics and more than  seven books, among them,  Elementary Number Theory with Applications, second edition; Catalan Numbers with Applications;  Triangular Arrays with Applications;and Discrete Mathematics with Applications.

Summary

Pell and Pell–Lucas numbers, like the well-known Fibonacci and Catalan numbers, continue to intrigue the mathematical world with their beauty and applicability. They offer opportunities for experimentation, exploration, conjecture, and problem-solving techniques, connecting the fields of analysis, geometry, trigonometry, and various areas of discrete mathematics, number theory, graph theory, linear algebra, and combinatorics. Pell and Pell–Lucas numbers belong to an extended Fibonacci family as a powerful tool for extracting numerous interesting properties of a vast array of number sequences.
A key feature of this work is the historical flavor that is interwoven into the extensive and in-depth coverage of the subject. An interesting array of applications to combinatorics, graph theory, geometry, and intriguing mathematical puzzles is another highlight engaging the reader. The exposition is user-friendly, yet rigorous, so that a broad audience consisting of students, math teachers and instructors, computer scientists and other professionals, along with the mathematically curious will all benefit from this book.
Finally, Pell and Pell–Lucas Numbers provides enjoyment and excitement while sharpening  the reader’s mathematical skills involving pattern recognition, proof-and-problem-solving techniques.​

Additional text

“The book under review contains a comprehensive overview of properties of Pell and Pell-Lucas numbers and their relations with a wealth of other mathematical objects. … The book contains many examples and most chapters end with a list of exercises, which makes the book particularly appropriate for undergraduate/graduate students.” (Wolfgang Steiner, zbMATH 1330.11002, 2016)
“The book is very appealing, it contains numerous figures and examples. And finally one thing is for sure – everyone will be able to find something interesting and new in this book, let be a student, a professor, or an amateur, who is enthralled by investigating integer sequences, and is willing to explore their hidden beauty.” (Péter Hajnal, Acta Scientiarum Mathematicarum, Vol. 82 (3-4), 2016)
“Books such as this support the pedagogical possibility for making ‘ontogeny recapitulate phylogeny’ for students learning mathematics in ways better mirroring how researchers first discovered it. … A thoroughreading will increase students’ knowledge of number theory (Diophantine equations, continued fractions), combinatorics (identities, generating functions, lattice paths, graphs), many families of related number sequences, and basic algebraic technique. A gold mine for undergraduate research prospects. Summing Up: Highly recommended. All readership levels.” (D. V. Feldman, Choice, Vol. 53 (1), September, 2015)
“One of the author’s main goals with this book is to ‘collect, organize, and present information about these integer families in a systematic and enjoyable fashion’. … The book is particularly appropriate for undergraduate/graduate students who are exploring the areas of combinatorics and number theory. It is also useful as a resource for research mathematicians as well as amateur mathematicians … . a delightful book written in an engaging style that draws the reader into taking the adventure of discovery.” (Edward G. Thurber, Mathematical Reviews, August, 2015)
“Thisbook can be used as a standalone or supplemental text in an upper level undergraduate, number-theory course. It could also be used as a supplemental text in a discrete mathematics course. Finally, it could also be read simply for its recreational flavor by a person in any field.” (Russell Jay Hendel, MAA Reviews, February, 2015)
“This is a treasure trove of relations, formulas, connections, that circle the notion of Pell numbers. There is no comparable publication having that amount of information available on this topic. It will be of great interest to number theorists, professional as well as amateurs.” (Adhemar Bultheel, euro-math-soc.au, January, 2015)

Report

"The book under review contains a comprehensive overview of properties of Pell and Pell-Lucas numbers and their relations with a wealth of other mathematical objects. ... The book contains many examples and most chapters end with a list of exercises, which makes the book particularly appropriate for undergraduate/graduate students." (Wolfgang Steiner, zbMATH 1330.11002, 2016)
"The book is very appealing, it contains numerous figures and examples. And finally one thing is for sure - everyone will be able to find something interesting and new in this book, let be a student, a professor, or an amateur, who is enthralled by investigating integer sequences, and is willing to explore their hidden beauty." (Péter Hajnal, Acta Scientiarum Mathematicarum, Vol. 82 (3-4), 2016)
"Books such as this support the pedagogical possibility for making 'ontogeny recapitulate phylogeny' for students learning mathematics in ways better mirroring how researchers first discovered it. ... A thoroughreading will increase students' knowledge of number theory (Diophantine equations, continued fractions), combinatorics (identities, generating functions, lattice paths, graphs), many families of related number sequences, and basic algebraic technique. A gold mine for undergraduate research prospects. Summing Up: Highly recommended. All readership levels." (D. V. Feldman, Choice, Vol. 53 (1), September, 2015)
"One of the author's main goals with this book is to 'collect, organize, and present information about these integer families in a systematic and enjoyable fashion'. ... The book is particularly appropriate for undergraduate/graduate students who are exploring the areas of combinatorics and number theory. It is also useful as a resource for research mathematicians as well as amateur mathematicians ... . a delightful book written in an engaging style that draws the reader into taking the adventure of discovery." (Edward G. Thurber, Mathematical Reviews, August, 2015)
"Thisbook can be used as a standalone or supplemental text in an upper level undergraduate, number-theory course. It could also be used as a supplemental text in a discrete mathematics course. Finally, it could also be read simply for its recreational flavor by a person in any field." (Russell Jay Hendel, MAA Reviews, February, 2015)
"This is a treasure trove of relations, formulas, connections, that circle the notion of Pell numbers. There is no comparable publication having that amount of information available on this topic. It will be of great interest to number theorists, professional as well as amateurs." (Adhemar Bultheel, euro-math-soc.au, January, 2015)

Product details

• Authors: Thomas Koshy
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 01.01.2016
• EAN: 9781493953417
• ISBN: 978-1-4939-5341-7
• No. of pages: 431
• Dimensions: 212 mm x 279 mm x 26 mm
• Weight: 1128 g
• Illustrations: XXIII, 431 p. 139 illus., 1 illus. in color.
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; B, History, Mathematics, Mathematics and Statistics, Number Theory, History of mathematics, History of Mathematical Sciences, Mathematical logic, Mathematical Logic and Foundations, Mathematical foundations