Fibonacci Numbers (Popular Lectures In Mathematics) | N. N. Vorob’ev

In elementary mathematics there are many difficult and interesting problems not connected with the name of an individual, but rather possessing the character of a kind of "mathematical folklore". Such problems are scattered throughout the wide literature of popular (or, simply, entertaining!) mathematics, and often it is very dif­ ficult to establish the source of a particular problem.

These problems often circulate in several versions. Sometimes several such problems combine into a single, more complex, one, sometimes the opposite happens and one problem splits up into several simple ones: thus it is often difficult to distinguish between the end of one

problem and the beginning of another. We should consider that in each of these problems we are dealing with little mathematical theories, each with its own history, its own complex of problems and its own characteristic methods, all, however, closely connected with the history and methods of "great mathematics".

The theory of Fibonacci numbers is just such a theory. Derived from the famous “rabbit problem”, going back nearly 750 years, Fibonacci numbers, even now, provide one of the most fascinating chapters of elementary mathe­ matics. Problems connected with Fibonacci numbers occur in many popular books on mathematics, are discussed at meetings of school mathematical societies, and feature in mathematical competitions.

The present booklet contains a set of problems which were the themes of several meetings of the school chil­dren’s mathematical club of Leningrad State University in the academic year 1949-50. In accordance with the wishes of those taking part, the questions discussed at these meetings were mostly number-theoretical, a theme which is developed in greater detail here.

This book is designed to appeal basically to pupils of 16 or 17 years of age in a high school. The concept o a limit is met with only in examples 7 and 8 in chapter III. The reader who is not acquainted with this concept can omit these without prejudice to his understanding of what follows. That applies also to binomial coefficients (I, example 8) and to trigonometry (IV, examples 2 & 3). The elements which are presented of the theory of divisi­bility and of the theory of continued fractions do not presuppose any knowledge beyond the limits of a school course.

Those readers who develop an interest in the principle of constructing recurrent series are recommended to read the small but full booklet of A.I. Markushevich, "Re­current Sequences" (Vozvratnyye posledovatel’ nosti) (Gostekhizdat, 1950). Those who become interested in facts relating to the theory of numbers are referred to textbooks in th is subject*.