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Fibonacci numbers arise in the analysis of the Fibonacci heap data structure. A one-dimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers. [74] The Fibonacci number series is used for optional lossy compression in the IFF 8SVX audio file format used on Amiga computers.
A repfigit, or Keith number, is an integer such that, when its digits start a Fibonacci sequence with that number of digits, the original number is eventually reached. An example is 47, because the Fibonacci sequence starting with 4 and 7 (4, 7, 11, 18, 29, 47) reaches 47.
In the Fibonacci sequence, each number is the sum of the previous two numbers. Fibonacci omitted the "0" and first "1" included today and began the sequence with 1, 2, 3, ... . He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377.
A prime divides if and only if p is congruent to ±1 modulo 5, and p divides + if and only if it is congruent to ±2 modulo 5. (For p = 5, F 5 = 5 so 5 divides F 5) . Fibonacci numbers that have a prime index p do not share any common divisors greater than 1 with the preceding Fibonacci numbers, due to the identity: [6]
144 (one hundred [and] forty-four) is the natural number following 143 and preceding 145.It is coincidentally both the square of twelve (a dozen dozens, or one gross.) and the twelfth Fibonacci number, and the only nontrivial number in the sequence that is square.
For generalized Fibonacci sequences (satisfying the same recurrence relation, but with other initial values, e.g. the Lucas numbers) the number of occurrences of 0 per cycle is 0, 1, 2, or 4. The ratio of the Pisano period of n and the number of zeros modulo n in the cycle gives the rank of apparition or Fibonacci entry point of n .
F, also called the Fibonacci factorial, where n is a nonnegative integer, is defined as the product of the first n positive Fibonacci numbers, i.e. !:= =,, where F i is the i th Fibonacci number, and 0! F gives the empty product (defined as the multiplicative identity, i.e. 1).
A page of the Liber Abaci from the National Central Library.The list on the right shows the numbers 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 (the Fibonacci sequence).