Fibonacci Numbers by Nicolai N. Vorobiev (auth.)

By Nicolai N. Vorobiev (auth.)

Since their discovery hundreds of thousands of years in the past, humans were serious about the wondrous houses of Fibonacci numbers. Being of mathematical importance of their personal correct, Fibonacci numbers have had an influence on components like paintings and structure, and their lines are available in nature or even the habit of the inventory industry. beginning with the elemental houses of Fibonacci numbers, the current ebook explores their relevance in quantity thought, the speculation of persisted fractions, geometry and approximation idea. instead of giving an entire account of the topic, a couple of selected examples are handled exhaustively. They not just show the bearing of Fibonacci numbers on arithmetic, but additionally offer very readable marvels of mathematical reasoning. This e-book is the interpretation of the sixth Russian version (the first version seemed within the early fifties and have become a typical resource of data at the subject).

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We conclude that d is a divisor of r3, ... , r n-1, and finally, of r n. Thus, we just proved that the Euclidean algorithm, when applied to two natural numbers a and b, does yield their greatest common divisor. The greatest common divisor of the numbers a and b will be denoted in what follows by (a, b). Clearly, a is divisible by b if and only if (a, b) = b. As an example, let us find (U20, U15) = (6765,610). 3) is 6765 = 610 . 11 + 55, 610 = 55· 11 + 5, 55 = 5·11. Thus, The fact that the greatest common divisor of two Fibonacci numbers turns out to be another Fibonacci number is not accidental.

Chapter 2 N umber-Theoretic Properties of Fibonacci Numbers 1. We are going to study some properties of Fibonacci numbers related to their divisibility by other numbers. The first result addresses the divisibility of a Fibonacci number by another Fibonacci number. Theorem. If n is divisible by m, then Un is divisible by Urn. Proof. Assume n is divisible by m and set n = mk. We will carry out the proof by induction on k. If k = 1, then n = m, and in this case it is obvious that Un is divisible by Urn.

However, the above mentioned drawback of the Fibonacci numeration sys- tem regarding the bigger capacity provided by the binary system turns out to be a positive feature of the system when it comes to what is called antijamming. Let us point this out as follows. In the binary system, any string of digits represents a certain number. Therefore, an error of any kind - omission or typo - in a string, produces a valid representation of a different number and, therefore, such an error would easily pass undetected.

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