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Solution to Fibonacci rabbit problem: In a growing idealized population, the number of rabbit pairs form the Fibonacci sequence. At the end of the n th month, the number of pairs is equal to F n. Relation to the golden ratio
The first part of Zeckendorf's theorem (existence) can be proven by induction. For n = 1, 2, 3 it is clearly true (as these are Fibonacci numbers), for n = 4 we have 4 = 3 + 1. If n is a Fibonacci number then there is nothing to prove. Otherwise there exists j such that F j < n < F j + 1 .
Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. [1] [2]Mathematical induction is a method for proving that a statement () is true for every natural number, that is, that the infinitely many cases (), (), (), (), … all hold.
A quick proof of Cassini's identity may be given (Knuth 1997, p. 81) by recognising the left side of the equation as a determinant of a 2×2 matrix of Fibonacci numbers. The result is almost immediate when the matrix is seen to be the n th power of a matrix with determinant −1:
In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind U n (P, Q) with relatively prime parameters P, Q and positive discriminant, an element U n with n ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12) = U 12 (1, − ...
Lamé's Theorem is the result of Gabriel Lamé's analysis of the complexity of the Euclidean algorithm.Using Fibonacci numbers, he proved in 1844 [1] [2] that when looking for the greatest common divisor (GCD) of two integers a and b, the algorithm finishes in at most 5k steps, where k is the number of digits (decimal) of b.
Although the resulting Fibonacci sequence dates back long before Leonardo, [9] its inclusion in his book is why the sequence is named after him today. The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots. [10] The book also includes proofs in Euclidean geometry. [11]
In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials .