Catalan's Identity

Theorem

Let $F_k$ be the $k$th Fibonacci number.

Then:

${F_n}^2 - F_{n - r} F_{n + r} = \left({-1}\right)^{n - r} {F_r}^2$

Proof 1

From the definition of Fibonacci numbers:

$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$
$F_n = F_{n - r} F_{r - 1} + F_{n - r + 1} F_r$

Also:

 $\ds F_{n + r}$ $=$ $\ds F_{n - r} F_{2 r - 1} + F_{n - r + 1} F_{2 r}$ Honsberger's Identity $\ds$ $=$ $\ds F_{n - r} \paren {F_{r - 1}^2 + {F_r}^2} + F_{n - r + 1} \paren {F_{r - 1} F_r + F_r F_{r + 1} }$ Honsberger's Identity $\ds$ $=$ $\ds F_{n - r} \paren { {F_{r - 1} }^2 + {F_r}^2} + F_{n - r + 1} \paren {F_{r - 1} F_r + F_r \paren {F_{r - 1} + F_r} }$ Definition of Fibonacci Number $\ds$ $=$ $\ds F_{n - r} \paren { {F_{r - 1} }^2 + {F_r}^2} + F_{n - r + 1} \paren {F_r \paren {F_{r - 1} + F_{r - 1} + F_r} }$ $\ds$ $=$ $\ds F_{n - r} \paren { {F_{r - 1} }^2 + {F_r}^2} + F_{n - r + 1} \paren {2 F_r F_{r - 1} + {F_r}^2}$

Therefore:

 $\ds$  $\ds F_n^2 - F_{n - r} F_{n + r}$ $\ds$ $=$ $\ds \paren {F_{n - r} F_{r - 1} + F_{n - r + 1} F_r}^2 - F_{n - r} \paren {F_{n - r} \paren { {F_{r - 1} }^2 + {F_r}^2} + F_{n - r + 1} \paren {2 F_r F_{r - 1} + {F_r}^2} }$ from above $\ds$ $=$ $\ds \paren { {F_{n - r + 1} }^2 {F_r}^2} - \paren { {F_{n - r} }^2 {F_r}^2 + F_{n - r} F_{n - r + 1} {F_r}^2}$ Expansion and Simplification $\ds$ $=$ $\ds {F_r}^2 \paren { {F_{n - r + 1} }^2 - {F_{n - r} }^2 - F_{n - r} F_{n - r + 1} }$ $\ds$ $=$ $\ds {F_r}^2 \paren {F_{n - r + 1} \paren {F_{n - r + 1} - F_{n - r} } - {F_{n - r} }^2}$ $\ds$ $=$ $\ds {F_r}^2 \paren {F_{n - r + 1} \paren {F_{n - r - 1} } - {F_{n-r} }^2}$ Definition of Fibonacci Number $\ds$ $=$ $\ds {F_r}^2 \paren {-1}^{n - r}$ Cassini's Identity $\ds$ $=$ $\ds \paren {-1}^{n - r} {F_r}^2$

$\blacksquare$

Proof 2

Proof by induction:

For all $n, r \in \N_{>0}$ where $n > r$, let $\map P {n, r}$ be the proposition:

${F_n}^2 - F_{n - r} F_{n + r} = \paren {-1}^{n - r} {F_r}^2$

Basis for the Induction

$n = 1$ yields no suitable $r$, so we look at $n = 2$ instead, which only gives us $r = 1$.

$\map P {2, 1}$ is true:

${F_2}^2 - F_3 F_1 = 1^2 - 2 \times 1 = -1 = -1 \times {F_1}^2$

$n = 3$ gives us only $r = 1$ and $r = 2$.

$\map P {3, 1}$ is true:

${F_3}^2 - F_2 F_4 = 2^2 - 1 \times 3 = 1 = 1 \times {F_1}^2$

$\map P {3, 2}$ is true:

${F_3}^2 - F_1 F_5 = 2^2 - 1 \times 5 = -1 = -1 \times {F_2}^2$

This is our basis for the induction.

First Induction Hypothesis

Now we need to show that, if $\map P {n, r}$ is true for all $r$, where $n > 3$, then it logically follows that $\map P {n + 1, r}$ is true for all $r$.

So this is our induction hypothesis:

$\forall r < n : {F_n}^2 - F_{n - r} F_{n + r} = \paren {-1}^{n - r} {F_r}^2$

Then we need to show:

$\forall r < n : {F_{n + 1} }^2 - F_{n - r + 1} F_{n + r + 1} = \paren {-1}^{n - r + 1} {F_r}^2$

Induction Step

This is our induction step:

It will again be a proof by induction.

Basis for the Induction

When $r = 1$:

 $\ds {F_{n + 1} }^2 - F_n F_{n + 2}$ $=$ $\ds {F_{n + 1} }^2 - F_n \paren {F_{n + 1} + F_n}$ Definition of Fibonacci Number $\ds$ $=$ $\ds {F_{n + 1} }^2 - F_n F_{n + 1} - {F_n}^2$ $\ds$ $=$ $\ds F_{n + 1} \paren {F_{n + 1} - F_n} - {F_n}^2$ $\ds$ $=$ $\ds F_{n + 1} F_{n - 1} - {F_n}^2$ Definition of Fibonacci Number $\ds$ $=$ $\ds \paren {-1} \paren {F_n^2 - F_{n - 1} F_{n + 1} }$ $\ds$ $=$ $\ds \paren {-1} \paren {-1}^{n - 1} {F_1}^2$ First induction hypothesis $\ds$ $=$ $\ds \paren {-1}^n {F_1}^2$

So $\map P {n + 1, 1}$ holds.

This is our basis for the induction.

Second Induction Hypothesis

Now we need to show that, if $\map P {n + 1, r}$ is true, where $2 < r < n$, then it logically follows that $\map P {n + 1, r + 1}$ is true.

So this is our second induction hypothesis:

${F_{n + 1} }^2 - F_{n - r + 1} F_{n + r + 1} = \paren {-1}^{n - r + 1} {F_r}^2$

Then we need to show:

${F_{n + 1} }^2 - F_{n - r} F_{n + r + 2} = \paren {-1}^{n - r} {F_{r + 1} }^2$

Induction Step

This is our induction step:

 $\ds {F_{n + 1} }^2 - F_{n - r} F_{n + r + 2}$ $=$ $\ds {F_{n + 1} }^2 - F_{n - r + 1} F_{n + r + 1} + F_{n - r + 1} F_{n + r + 1} - F_{n - r} F_{n + r + 2}$ $\ds$ $=$ $\ds \paren {-1}^{n - r + 1} {F_r}^2 + F_{n - r + 1} F_{n + r + 1} - F_{n - r} F_{n + r + 2}$ Second induction hypothesis $\ds$ $=$ $\ds \paren {-1}^{n - r + 1} {F_r}^2 + F_{n - r + 1} F_{n + r + 1} - F_{n - r} \paren {F_{n + r} + F_{n + r + 1} }$ Definition of Fibonacci Number $\ds$ $=$ $\ds \paren {-1}^{n - r + 1} {F_r}^2 + \paren {F_{n - r + 1} - F_{n - r} } F_{n + r + 1} - F_{n - r} F_{n + r}$ $\ds$ $=$ $\ds \paren {-1}^{n - r + 1} {F_r}^2 + F_{n - r - 1} F_{n + r + 1} - F_{n - r} F_{n + r}$ Definition of Fibonacci Number $\ds$ $=$ $\ds \paren {-1}^{n - r + 1} {F_r}^2 - F_{n - r} F_{n + r} + F_{n - r - 1} F_{n + r + 1}$ $\ds$ $=$ $\ds \paren {-1}^{n - r + 1} {F_r}^2 + {F_n}^2 - F_{n - r} F_{n + r} - {F_n}^2 + F_{n - r - 1} F_{n + r + 1}$ $\ds$ $=$ $\ds \paren {-1}^{n - r + 1} {F_r}^2 + \paren {-1}^{n - r} {F_r}^2 - \paren {-1}^{n - r + 1} {F_{r + 1} }^2$ First induction hypothesis $\ds$ $=$ $\ds \paren {-1}^{n - r + 1} {F_r}^2 + \paren {-1}^{n - r} {F_r}^2 + \paren {-1}^{n - r} {F_{r + 1} }^2$ $\ds$ $=$ $\ds \paren {-1}^{n - r} \paren {- {F_r}^2 + {F_r}^2} + \paren {-1}^{n - r} {F_{r + 1} }^2$ $\ds$ $=$ $\ds \paren {-1}^{n - r} {F_{r + 1} }^2$

So $\map P {n + 1, r} \implies \map P {n + 1, r + 1}$ and the result follows by the Principle of Mathematical Induction.

So $\map P {n, r} \implies \map P {n + 1, r}$ and the result follows by the Principle of Mathematical Induction.

Therefore:

$F_n^2 - F_{n - r} F_{n + r} = \paren {-1}^{n - r} {F_r}^2$

$\blacksquare$

Source of Name

This entry was named for Eugène Charles Catalan.