Samson identity

The Cassini identity or Simson identity describes a relationship between three consecutive Fibonacci numbers . It is named after Giovanni Domenico Cassini (1625–1712) or Robert Simson (1687–1768), which they proved independently of one another, and also a special case of the more general identity of Catalan .

Identity and Generalizations

For three consecutive Fibonacci numbers with : ${\displaystyle f_{n-1},f_{n},f_{n+1}}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle f_{n-1}f_{n+1}-f_{n}^{2}=(-1)^{n}}$

This is a special case of Catalan's identity ( ) ${\displaystyle n,k\in \mathbb {N} ,\,n>k}$

${\displaystyle f_{n}^{2}-f_{n-k}f_{n+k}=(-1)^{n-k}f_{k}^{2}}$,

which in turn can be generalized to the identity of Vadja ( ): ${\displaystyle n,i,j\in \mathbb {N} }$

${\displaystyle f_{n+i}f_{n+j}-f_{n}f_{n+i+j}=(-1)^{n}f_{i}f_{j}}$.

proof

A very brief proof of Cassini's identity results from the matrix representation of the Fibonacci numbers:

${\displaystyle f_{n-1}f_{n+1}-f_{n}^{2}=\det \left(\left[{\begin{matrix}f_{n+1}&f_{n}\\f_{n}&f_{n-1}\end{matrix}}\right]\right)=\det \left(\left[{\begin{matrix}1&1\\1&0\end{matrix}}\right]^{n}\right)=\det \left(\left[{\begin{matrix}1&1\\1&0\end{matrix}}\right]\right)^{n}=(-1)^{n}}$

history

The French astronomer and mathematician Cassini proved the identity in 1680 and the Scottish mathematician Simson independently of it in 1753. However, the identity was probably already known to Johannes Kepler around 1608. The Belgian mathematician Eugène Charles Catalan (1814–1894) published the identity named after him in 1879. The British mathematician Steven Vajda (1901–95) wrote a book on Fibonacci numbers ( Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications , 1989 ), which contains the identity named after him. However, this identity was published by Dustan Everman in The American Mathematical Monthly as early as 1960 .

literature

• Thomas Koshy: Fibonacci and Lucas Numbers with Applications . Wiley, 2001, ISBN 9781118031315 , pp. 74-75, 83, 88
• Albrecht Beutelspacher , Bernhard Petri: The golden ratio. Spectrum, Heidelberg / Berlin / Oxford 1996. ISBN 3-86025-404-9 , pp. 91-93

Web links

• Eric W. Weisstein : Cassini's Identity . In: MathWorld (English).
• Cassini's Identity on cut-the-knot.org
• Cassini's Identity in the ProofWiki

Individual evidence

1. ^ Albrecht Beutelspacher , Bernhard Petri: The golden ratio. Spectrum, Heidelberg / Berlin / Oxford 1996. ISBN 3-86025-404-9 , pp. 91-93
2. ↑ ^{a b c d} Thomas Koshy: Fibonacci and Lucas Numbers with Applications . Wiley, 2001, ISBN 9781118031315 , pp. 74-75, 83, 88
3. ^ ^{A b} Douglas B. West: Combinatorial Mathematics . Cambridge University Press, 2020, p. 61
4. ^ Donald Ervin Knuth : The Art of Computer Programming, Volume 1: Fundamental Algorithms . Addison-Wesley, ISBN 0-201-89683-4 , p. 81
5. ^ Miodrag Petkovic: Famous Puzzles of Great Mathematicians . AMS, 2009, ISBN 9780821848142 , pp. 30-31
6. ↑ Steven Vadja: Fibonacci and Lucas Numbers and the Golden Section: Theory and Applications . Dover, 2008, ISBN 978-0486462769 , p. 28 (first published in 1989 by Ellis Horwood)