Formula F 0-1-in-2
In other words,. This is the same as requiring a and b satisfy the system of equations:. Taking the starting values U 0 and U 1 to be arbitrary constants, a more general solution is:. Therefore, it can be found by rounding , that is by the use of the nearest integer function:. Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, , , , , The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio. The resulting recurrence relationships yield Fibonacci numbers as the linear coefficients:.
This equation can be proved by induction on n. A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is. From this, the n th element in the Fibonacci series may be read off directly as a closed-form expression :.
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Equivalently, the same computation may performed by diagonalization of A through use of its eigendecomposition :. This property can be understood in terms of the continued fraction representation for the golden ratio:. The matrix representation gives the following closed-form expression for the Fibonacci numbers:.
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Taking the determinant of both sides of this equation yields Cassini's identity ,. This matches the time for computing the n th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number recursion with memoization. The question may arise whether a positive integer x is a Fibonacci number.
This formula must return an integer for all n , so the radical expression must be an integer otherwise the logarithm does not even return a rational number. Here, the order of the summand matters. One group contains those sums whose first term is 1 and the other those sums whose first term is 2. It follows that the ordinary generating function of the Fibonacci sequence, i. Numerous other identities can be derived using various methods. Some of the most noteworthy are: . The last is an identity for doubling n ; other identities of this type are.
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These can be found experimentally using lattice reduction , and are useful in setting up the special number field sieve to factorize a Fibonacci number. More generally, . The generating function of the Fibonacci sequence is the power series. This can be proved by using the Fibonacci recurrence to expand each coefficient in the infinite sum:. In particular, if k is an integer greater than 1, then this series converges.
Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, we can write the sum of every odd-indexed reciprocal Fibonacci number as. No closed formula for the reciprocal Fibonacci constant. The Millin series gives the identity . Every third number of the sequence is even and more generally, every k th number of the sequence is a multiple of F k. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property  .
Any three consecutive Fibonacci numbers are pairwise coprime , which means that, for every n ,. These cases can be combined into a single formula, using the Legendre symbol : .
If n is composite and satisfies the formula, then n is a Fibonacci pseudoprime. When m is large—say a bit number—then we can calculate F m mod n efficiently using the matrix form. Here the matrix power A m is calculated using modular exponentiation , which can be adapted to matrices.
A Fibonacci prime is a Fibonacci number that is prime. The first few are:. Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many. As there are arbitrarily long runs of composite numbers , there are therefore also arbitrarily long runs of composite Fibonacci numbers. The only nontrivial square Fibonacci number is Bugeaud, M.
Mignotte, and S.
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Siksek proved that 8 and are the only such non-trivial perfect powers. No Fibonacci number can be a perfect number. Such primes if there are any would be called Wall—Sun—Sun primes. For odd n , all odd prime divisors of F n are congruent to 1 modulo 4, implying that all odd divisors of F n as the products of odd prime divisors are congruent to 1 modulo 4.
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Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field. However, for any particular n , the Pisano period may be found as an instance of cycle detection.
Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple.
The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle. The first triangle in this series has sides of length 5, 4, and 3.
This series continues indefinitely. The triangle sides a , b , c can be calculated directly:. The Fibonacci sequence is one of the simplest and earliest known sequences defined by a recurrence relation , and specifically by a linear difference equation.
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All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients. From Wikipedia, the free encyclopedia. For the chamber ensemble, see Fibonacci Sequence ensemble. Further information: Patterns in nature.
Main article: Golden ratio. Main article: Cassini and Catalan identities. Main article: Fibonacci prime. Main article: Pisano period. Main article: Generalizations of Fibonacci numbers. Wythoff array. In this way, for six, [variations] of four [and] of five being mixed, thirteen happens.