Fibonacci numberIn the text Liber Abaci written by Fibonacci in 1202, the following question was posed.
A man puts one pair of rabbits in a certain place entirely surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year, if the nature of these rabbits is such that every month each pair bears a new pair which from the second month becomes productive?The number of pairs of rabbits in the nth month begins 1, 1, 2, 3, 5, 8, 13, 21, ... (often denoted u1, u2, u3 ...) where each term is the sum of the two terms preceding it. Mathematicians define this sequence recursively as follows:
u1 = u2 = 1 and un+1 = un + un-1 (n > 2)This sequence, now called the Fibonacci sequence, has an amazing number of applications in nature and art; it also has a tremendous number of interesting properties--which is reason enough for the journal "The Fibonacci Quarterly" to exist! Here we list just a few of these many properties. Let m and n be positive integers,
- un divides umn,
- gcd(un, um) = ugcd(m,n),
- un2 - un+1un-1 = (-1)n-1,
- u1 + u3 + u5 + ... + u2n-1 = u2n,
- for every n, there are n consecutive composite Fibonacci numbers,
- every positive integer can be written as a sum of distinct Fibonacci numbers, and
- the product of any four consecutive Fibonacci numbers is the area of a Pythagorean triangle.
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