Math 2.0

Number theory Theorem

Cassini's identity

Builds on Fibonacci sequence

Statement

Let nn be a positive integer, and let FmF_m be the Fibonacci numbers. Then Fn−1Fn+1−Fn2=(−1)n.F_{n-1} F_{n+1} - F_n^2 = (-1)^n.

Remarks

Examples: for n=5n = 5, F4F6−F52=3⋅8−25=−1F_4 F_6 - F_5^2 = 3 \cdot 8 - 25 = -1. For n=6n = 6, F5F7−F62=5⋅13−64=1F_5 F_7 - F_6^2 = 5 \cdot 13 - 64 = 1. A short proof takes determinants in the matrix identity (1110)n=(Fn+1FnFnFn−1)\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}. Catalan's identity Fn2−Fn−rFn+r=(−1)n−rFr2F_n^2 - F_{n-r} F_{n+r} = (-1)^{n-r} F_r^2 generalizes it. The case r=1r = 1 is Cassini's identity.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

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Fibonacci sequence, example, no evidence levelFibonacci sequenceexampleCassini's identity, statement, no evidence levelCassini's identitystatement
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  1. Fibonacci sequence example
  2. Cassini's identity statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cassini_and_Catalan_identities, lead section
About this page
Tag01WQZK97Z5
URIm2://01WQZK97Z5
Kindstatement, theorem
WritingW1 stated
MSC11B39
StatusThis copy of the site has no build data. Run m2 status 01WQZK97Z5.
UsesFibonacci sequence

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