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**Fibonacci and Lucas numbers with applications. Volume I.**
*(English)*
Zbl 0984.11010

The most noticeable feature of this book is its length (652 pages, 47 chapters). The author has attempted to appeal to a very wide audience: from high school students to researchers. In so doing, he has created a hodge-podge. The book resembles a textbook inasmuch as (1) all chapters starting with the second end with numerous exercises; (2) solutions to odd-numbered exercises appear at length at the end of the book: (3) some chapters are devoted to items from undergraduate mathematics.

As a reference material, this book falls somewhat short of the mark. Although there are numerous references to research results, these references leave something to be desired, since they do not follow the customary practice of being numbered. In addition, some research results are undocumented. (For example, see page 200, result by Cross and Renzi.)

The book’s many illustrations and photographs are informative and esthetic. On the other hand, the author sometimes gets carried away in his prose, referring to the Fibonacci and Lucas sequences as “twin stars”. Another source of annoyance is the author’s somewhat slipshod historical scholarship. Specifically, there is an undocumented statement about Fibonacci’s business trips abroad, confusion of the Roman Emperor with the Holy Roman Emperor, and a dubious reference to Kepler.

In summary, if you wish to learn something about Fibonacci numbers, you may consult this book, but you may also wish to consult prior works on the same subject by N. N. Vorobiev [Fibonacci numbers. Basel: BirkhĂ¤user (2002; Zbl 1014.11012)], V. E. Hoggatt jun. [Fibonacci and Lucas numbers. Boston etc.: Houghton Mifflin Company (1969; Zbl 0198.36903)], and S. Vajda [Fibonacci and Lucas numbers, and the golden section. Chichester: Ellis Horwood Ltd. etc. (1989; Zbl 0695.10001)].

As a reference material, this book falls somewhat short of the mark. Although there are numerous references to research results, these references leave something to be desired, since they do not follow the customary practice of being numbered. In addition, some research results are undocumented. (For example, see page 200, result by Cross and Renzi.)

The book’s many illustrations and photographs are informative and esthetic. On the other hand, the author sometimes gets carried away in his prose, referring to the Fibonacci and Lucas sequences as “twin stars”. Another source of annoyance is the author’s somewhat slipshod historical scholarship. Specifically, there is an undocumented statement about Fibonacci’s business trips abroad, confusion of the Roman Emperor with the Holy Roman Emperor, and a dubious reference to Kepler.

In summary, if you wish to learn something about Fibonacci numbers, you may consult this book, but you may also wish to consult prior works on the same subject by N. N. Vorobiev [Fibonacci numbers. Basel: BirkhĂ¤user (2002; Zbl 1014.11012)], V. E. Hoggatt jun. [Fibonacci and Lucas numbers. Boston etc.: Houghton Mifflin Company (1969; Zbl 0198.36903)], and S. Vajda [Fibonacci and Lucas numbers, and the golden section. Chichester: Ellis Horwood Ltd. etc. (1989; Zbl 0695.10001)].

Reviewer: Neville Robbins (San Francisco)

### MSC:

11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

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\textit{T. Koshy}, Fibonacci and Lucas numbers with applications. Volume I. New York, NY: Wiley (2001; Zbl 0984.11010)

### Online Encyclopedia of Integer Sequences:

Squares of Lucas numbers.Expansion of (1+x^2)/(1 - 2*x - 2*x^2 + x^3).

Fermat’s Diophantine m-tuple: 1 + the product of any two distinct terms is a square.

Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.

A Fibonacci triangle: triangle T(n,k) = Fibonacci(k+1)*Fibonacci(n-k+1), for n >= 0, 0 <= k <= n.

a(n) = phi(Fibonacci(n)).

Powers of Lucas numbers.

Numbers n such that 5*n + 2 is prime.

Nonprime Lucas numbers.

Lucas multiplication table as a triangle read by rows T(n, k) = L(n)L(k), with L(n) being a Lucas number (A000032).

Odd Fibonacci numbers with odd index.

(5*F(n)+3*L(n)-8)/2.

Integers nearest to (2^((n-3)/2) + 3^((n-3)/2)).

a(n) = floor(sqrt(F(n+2)^2 + F(n)^2)), where F(n) = A000045(n).