Math 2.0

Number theory Theorem

Addition formula for Fibonacci numbers

Builds on Fibonacci sequence

Statement

Let mm and nn be natural numbers, and let FkF_k be the Fibonacci numbers. Then Fm+n+1=FmFn+Fm+1Fn+1.F_{m+n+1} = F_m F_n + F_{m+1} F_{n+1}.

Remarks

With m=nm = n it gives F2n+1=Fn2+Fn+12F_{2n+1} = F_n^2 + F_{n+1}^2. An equivalent form is Fm+n=Fm−1Fn+FmFn+1F_{m+n} = F_{m-1} F_n + F_m F_{n+1} for m≥1m \ge 1. The formula is the basis of fast algorithms that compute FnF_n with O(log⁡n)O(\log n) arithmetic operations.

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  1. Fibonacci sequence example
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Further reading

  1. Web https://en.wikipedia.org/wiki/Fibonacci_sequence, Other identities
About this page
Tag00CN2F997Q
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Kindstatement, theorem
WritingW0 stub
MSC11B39
StatusThis copy of the site has no build data. Run m2 status 00CN2F997Q.
UsesFibonacci sequence

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