Edouard Lucas – the generalisation of Fibonacci numbers
The French mathematician Edouard Lucas (1842–1891) studied in detail sequences of positive numbers where each element is the sum of the previous two and the initial two elements are chosen arbitrarily. The most famous sequence of this type is the one Fibonacci introduced with his rabbit problem. It was Lucas who acknowledged Leonardo of Pisa as the legitimate discoverer of this sequence, after other scientists such as Gabriel Lamé had claimed to have discovered the sequence. Since then, the terms "Fibonacci sequence" and "Fibonacci numbers" are commonly used.
Relationships between Fibonacci and Lucas numbers
Lucas also established his own sequence of numbers, known as the "Lucas sequence": 1, 3, 4, 7, 11, 18, 29, 47 … The sequence of Lucas numbers is defined in accordance with the same principle as Fibonacci numbers: L_n = L_(n−1) + L_(n−2). Here, however, the starting values are L_1 = 1 and L_2 = 3. There are countless connections between Fibonacci and Lucas numbers. For example, the addition of specific Fibonacci numbers yields a Lucas number: F_1 + F_3 = L_2, F_2 + F_4 = L_3, etc. In generalised form: L_n = F_(n−1) + F_(n+1)
Periodicity of last digits
Both number sequences show striking periodic repetitions of last digits. With Fibonacci numbers, the sequence of last digits recurs after a period of 60. After a cycle of 300, the last two digits repeat, after a cycle of 1,500 the last three digits, after a cycle of 15,000 the last four digits and so on. With Lucas numbers, the last digits recur with a period of 12. The last two digits are repeated after a cycle of 60, the last three after a cycle of 300, the last four after a cycle of 3,000 and so on.
Another phenomenon is that, with both sequences, the ratios of consecutive values tend to the golden ratio.
Literature on Lucas numbers
- Verner E. Hoggatt, Fibonacci and Lucas Numbers. New York, 1969
- Edouard Lucas, Considérations nouvelles sur la théorie des nombres premiers et sur la division géométrique de la circonférence en parties égales. Association française pour l'avancement des sciences, 6 (1877), pp. 159–167.