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5.3 Induction proofs. 5.4 Binet ... the Fibonacci sequence is a sequence in which each element is ... The above formula can be used as a primality test in the ...
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 .
In 370 BC, Plato's Parmenides may have contained traces of an early example of an implicit inductive proof, [5] however, the earliest implicit proof by mathematical induction was written by al-Karaji around 1000 AD, who applied it to arithmetic sequences to prove the binomial theorem and properties of Pascal's triangle.
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:
A Fibonacci sequence of order n is an integer sequence in which each sequence element is the sum of the previous elements (with the exception of the first elements in the sequence). The usual Fibonacci numbers are a Fibonacci sequence of order 2.
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, − ...
[15] [16] However, even Fibonacci pseudoprimes do exist (sequence A141137 in the OEIS) under the first definition given by . A strong Fibonacci pseudoprime is a composite number n for which congruence holds for Q = −1 and all P. [17] It follows [17]: 460 that an odd composite integer n is a strong Fibonacci pseudoprime if and only if:
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]