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Pell and Pell-Lucas Numbers with Applications

Thomas Koshy
Publication Date: 
Number of Pages: 
[Reviewed by
Russell Jay Hendel
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This book 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.

  • In a number theory course: The book presents summation notation, product notation, basic congruences, recursions, Diophantine equations (Pythagorean triplets and Pell's equation), and continued fractions, standard topics in a number theory course. However, the book does not present quadratic reciprocity or the number-theoretic functions such as the phi and totient functions, also standard topics in a first year number theory course. Thus the book could either be used as a 2nd text, or, if the instructor is more ambitious, it could be used as a standalone text for a non-traditional number theory course. This is appealing since many interesting topics, not normally covered in number theory courses are presented, including generating functions, matrix applications, Chebychev polynomials, relations to trigonometric functions, and applications of graphing and tiling. Several recreational topics such as triangular numbers are covered in depth.
  • In a discrete mathematics course: The book contains several topics important in a discrete mathematics course including recursions, generating functions, matrix applications and use of graphs and tilings.
  • Read for recreation: There are certain books that any mathematician can read to keep abreast of other fields and for purposes of pure enjoyment. This book with its lively style, slow-paced proofs, historical tidbits and variety of applications has such a recreational flavor.

The book has several desirable features of a good textbook.

  • Prerequisites: The book is extremely student friendly. It even reviews elementary topics such as the binomial expansion, summation, product notation and matrices.
  • Coverage: The book has 20 chapters and about 200 sections. Thus the instructor has a wide range of topics to chose from
  • Exercises: Many, but not all, chapters have exercises. There are 275 exercises. They come in a variety of challenge levels.
  • Historical tidbits: The book is full of delicious vignettes. For example, did you know that Pell had almost nothing to do with Pell’s equation? The problem was first posed by Fermat to Wallis and Brouncker. And it was Euler who erroneously transferred credit to Pell.
  • Breadth: This is a particularly strong point of the book. Pell-Lucas sounds like a topic in discrete mathematics. But the book covers a variety of advanced topics such as Chebychev polynomials, graphs, tilings, and connections with trigonometric functions. The book also presents unsolved problems.
  • References: This is another strong point of the book. The author has skillfully woven 269 references, many obscure, into a cohesive and well organized book. The availability of such a wealth of references is particularly suited to undergraduate courses and seminars since undergraduates can be encouraged to read (and present!) original papers in class.

Finally, this reviewer was most pleased to see three of his publications cited. This shows the thoroughness of the author in his preparatory research for the book.

Russell Jay Hendel (RHendel@Towson.Edu) holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.

List of Symbols

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