__notoc__ Cassini's identity (sometimes called Simson's identity) and Catalan's identity are mathematical identities for the

Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...

s. Cassini's identity, a special case of Catalan's identity, states that for the ''n''th Fibonacci number, :

F_F_ - F_n^2 = (-1)^n.

Note here

F_0

is taken to be 0, and

F_1

is taken to be 1. Catalan's identity generalizes this: :

F_n^2 - F_F_ = (-1)^F_r^2.

Vajda's identity generalizes this: :

F_F_ - F_F_ = (-1)^nF_F_.

History

Cassini's formula was discovered in 1680 by

Giovanni Domenico Cassini Giovanni Domenico Cassini, also known as Jean-Dominique Cassini (8 June 1625 – 14 September 1712) was an Italian (naturalised French) mathematician, astronomer and engineer. Cassini was born in Perinaldo, near Imperia, at that time in the ...

, then director of the Paris Observatory, and independently proven by

Robert Simson Robert Simson (14 October 1687 – 1 October 1768) was a Scottish mathematician and professor of mathematics at the University of Glasgow. The Simson line is named after him. However Johannes Kepler presumably knew the identity already in 1608.

Eugène Charles Catalan Eugène Charles Catalan (30 May 1814 – 14 February 1894) was a French and Belgian mathematician who worked on continued fractions, descriptive geometry, number theory and combinatorics. His notable contributions included discovering a periodic ...

found the identity named after him in 1879. The British mathematician Steven Vajda (1901–95) published a book on Fibonacci numbers (''Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications'', 1989) which contains the identity carrying his name.Douglas B. West: ''Combinatorial Mathematics''. Cambridge University Press, 2020, p
61
/ref>Steven Vadja: ''Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications''. Dover, 2008, , p. 28 (original publication 1989 at Ellis Horwood) However the identity was already published in 1960 by Dustan Everman as problem 1396 in The American Mathematical Monthly.Thomas Koshy: ''Fibonacci and Lucas Numbers with Applications''. Wiley, 2001, , pp. 74-75, 83, 88

Proof of Cassini identity

Proof by matrix theory

A quick proof of Cassini's identity may be given by recognising the left side of the equation as a

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...

of a 2×2

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

of Fibonacci numbers. The result is almost immediate when the matrix is seen to be the th power of a matrix with determinant −1: :

F_F_ - F_n^2
=\det\left beginF_&F_n\\F_n&F_\end\right =\det\left begin1&1\\1&0\end\right n
=\left(\det\left begin1&1\\1&0\end\right right)^n
=(-1)^n.

Proof by induction

Consider the induction statement: :

F_F_ - F_n^2 = (-1)^n

The base case

n=1

is true. Assume the statement is true for

n

. Then: :

F_F_ - F_n^2 + F_nF_ - F_nF_ = (-1)^n

F_F_ + F_nF_ - F_n^2 - F_nF_ = (-1)^n

F_(F_ + F_n) - F_n(F_n + F_) = (-1)^n

F_^2 - F_nF_ = (-1)^n

F_nF_ - F_^2 = (-1)^

so the statement is true for all integers

n>0

Proof of Catalan identity

We use

Binet's formula In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...

, that

F_n=\frac

, where

\phi=\frac

and

\psi=\frac

. Hence,

\phi+\psi=1

and

\phi\psi=-1

. So, :

5(F_n^2 - F_F_)

= (\phi^n-\psi^n)^2 - (\phi^-\psi^)(\phi^-\psi^)

= (\phi^ - 2\phi^\psi^ +\psi^) - (\phi^ - \phi^\psi^(\phi^\psi^+\phi^\psi^) + \psi^)

= - 2\phi^\psi^ + \phi^\psi^(\phi^\psi^+\phi^\psi^)

Using

\phi\psi=-1

, :

= -(-1)^n2 + (-1)^n(\phi^\psi^+\phi^\psi^)

and again as

\phi=\frac

, :

= -(-1)^n2 + (-1)^(\psi^+\phi^)

The

Lucas number The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci n ...

L_n

is defined as

L_n=\phi^n+\psi^n

, so :

= -(-1)^n2 + (-1)^L_

Because

L_ = 5 F_n^2 + 2(-1)^n

= -(-1)^n2 + (-1)^(5 F_r^2 + 2(-1)^r)

= -(-1)^n2 + (-1)^2(-1)^r + (-1)^5 F_r^2

= -(-1)^n2 + (-1)^n2 + (-1)^5 F_r^2

= (-1)^5 F_r^2

Cancelling the

5

's gives the result.

Notes

References

* * *{{cite journal , last1 = Werman , first1 = M. , authorlink2 = Doron Zeilberger , last2 = Zeilberger , first2 = D. , title = A bijective proof of Cassini's Fibonacci identity , journal = Discrete Mathematics , volume = 58 , issue = 1 , year = 1986 , pages = 109 , mr = 0820846 , doi = 10.1016/0012-365X(86)90194-9, doi-access = free

External links

Proof of Cassini's identityProof of Catalan's Identity
Mathematical identities Fibonacci numbers Articles containing proofs Giovanni Domenico Cassini