The sequence of numbers defined by the in the Lucas Sequence. They are companions to the Lucas
Numbers and satisfy the same Recurrence Relation,

(1) |

The ratios of alternate Fibonacci numbers are given by the convergents to , where is the Golden Ratio, and are said to measure the fraction of a turn between successive leaves on the stalk of a plant (Phyllotaxis): 1/2 for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. (Coxeter 1969, Ball and Coxeter 1987). The Fibonacci numbers are sometimes called Pine Cone Numbers (Pappas 1989, p. 224).

Another Recurrence Relation for the Fibonacci numbers is

(2) |

(3) |

The Generating Function for the Fibonacci numbers is

(4) |

Yuri Matijasevic (1970) proved that the equation is a Diophantine Equation. This led to the proof of the impossibility of the tenth of Hilbert's Problems (does there exist a general method for solving Diophantine Equations?) by Julia Robinson and Martin Davis in 1970.

The Fibonacci number gives the number of ways for Dominoes to cover a Checkerboard, as illustrated in the following diagrams (Dickau).

The number of ways of picking a Set (including the Empty Set) from the numbers 1, 2, ..., without picking two consecutive numbers is . The number of ways of picking a set (including the Empty Set) from the numbers 1, 2, ..., without picking two consecutive numbers (where 1 and are now consecutive) is , where is a Lucas Number. The probability of not getting two heads in a row in tosses of a Coin is (Honsberger 1985, pp. 120-122). Fibonacci numbers are also related to the number of ways in which Coin Tosses can be made such that there are not three consecutive heads or tails. The number of ideals of an -element Fence Poset is the Fibonacci number .

Sum identities are

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

(16) |

(17) |

(18) |

(19) |

(20) |

(21) |

(22) |

(23) |

In terms of the Lucas Number ,

(24) |

(25) |

(26) |

(27) |

(28) |

(29) |

(30) |

(31) |

The equation (1) is a Linear Recurrence Sequence

(32) |

(33) |

(34) |

(35) |

(36) |

(37) |

From (1), the Ratio of consecutive terms is

(38) |

which is just the first few terms of the Continued Fraction for the Golden Ratio . Therefore,

(39) |

The ``Shallow Diagonals'' of Pascal's Triangle sum to Fibonacci numbers (Pappas 1989),

(40) |

where is a Generalized Hypergeometric Function.

The sequence of final digits in Fibonacci numbers repeats in cycles of 60. The last two digits repeat in 300, the last
three in 1500, the last four in 15,000, etc.

(41) |

which is equal to the fractional digits of ,

(42) |

For , Iff . Iff divides into an Even number of times. (Michael 1964; Honsberger 1985, pp. 131-132). No Odd Fibonacci number is divisible by 17 (Honsberger 1985, pp. 132 and 242). No Fibonacci number is ever of the form or where is a Prime number (Honsberger 1985, p. 133).

Consider the sum

(43) |

(44) |

(45) |

(46) |

(47) | |||

(48) |

so

(49) |

(50) |

(51) |

The Fibonacci numbers are Complete. In fact, dropping one number still leaves a Complete
Sequence, although dropping two numbers does not (Honsberger 1985, pp. 123 and 126). Dropping two terms from the Fibonacci
numbers produces a sequence which is not even Weakly Complete (Honsberger 1985, p. 128).
However, the sequence

(52) |

For a discussion of Square Fibonacci numbers, see Cohn (1964), who proved that the only
Square Number Fibonacci numbers are 1 and (Cohn 1964, Guy 1994). Ming (1989) proved that the only
Triangular Fibonacci numbers are 1, 3, 21, and 55. The Fibonacci and Lucas
Numbers have no common terms except 1 and 3. The only Cubic Fibonacci numbers are 1 and 8.

(53) |

(54) |

In 1975, James P. Jones showed that the Fibonacci numbers are the Positive Integer values of the Polynomial

(55) |

Every that is Prime has a Prime index , with the exception of . However, the converse is not true (i.e., not every prime index gives a Prime ). The first few Prime Fibonacci numbers are for , 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, ... (Sloane's A001605; Dubner and Keller 1999). Gardner's statement that is prime is incorrect, especially since 531 is not even Prime (Gardner 1979, p. 161). It is not known if there are an Infinite number of Fibonacci primes.

The Fibonacci numbers , are Squareful for , 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, ..., 372, 375, 378, 384, ... (Sloane's A037917) and Squarefree for , 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... (Sloane's A037918). and for all , but no Squareful Fibonacci numbers are known with Prime.

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© 1996-9

1999-05-26