Pell Number as Sum of Squares

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Theorem

Let $P_n$ be a Pell Number:

$P_n = \begin{cases} 0 & : n = 0 \\

1 & : n = 1 \\ 2 P_{n - 1} + P_{n - 2} & : \text {otherwise}\end{cases}$

Then:

$P_{2 n + 1} = P_{n + 1}^2 + P_n^2$


Proof

This proof proceeds by induction.


Basis for the Induction

$\map P 0$ is the case:

\(\ds P_{2 \times 0 + 1}\) \(=\) \(\ds P_{0 + 1}^2 + P_0^2\)
\(\ds 1\) \(=\) \(\ds 1^2 + 0^2\)

So $\map P 0$ is seen to hold.


$\map P 1$ is the case:

\(\ds P_{2 \times 1 + 1}\) \(=\) \(\ds P_{1 + 1}^2 + P_1^2\)
\(\ds 5\) \(=\) \(\ds 2^2 + 1^2\)

So $\map P 1$ is seen to hold.


Induction Hypothesis

Now we need to show that, if $\map P {n - 1}$ and $\map P n$ are true, where $n \ge 1$, then it logically follows that $\map P {n + 1}$ is true.


So this is our induction hypotheses:

$P_{2 n - 1} = P_n^2 + P_{n - 1}^2$
$P_{2 n + 1} = P_{n + 1}^2 + P_n^2$

from which we are to show:

$P_{2 n + 3} = P_{n + 2}^2 + P_{n + 1}^2$


Induction Step

This is our induction step:

\(\ds P_{2 n + 3}\) \(=\) \(\ds 2 P_{2 n + 2} + P_{2 n + 1}\) Definition of Pell Numbers
\(\ds \) \(=\) \(\ds 2 \paren {2 P_{2 n + 1} + P_{2 n} } + P_{2 n + 1}\) Definition of Pell Numbers
\(\ds \) \(=\) \(\ds 5 P_{2 n + 1} + 2 P_{2 n}\)
\(\ds \) \(=\) \(\ds 5 \paren {P_{n + 1}^2 + P_n^2} + \paren {P_{2 n + 1} - P_{2 n - 1} }\) induction hypothesis and Definition of Pell Numbers
\(\ds \) \(=\) \(\ds 5 \paren {P_{n + 1}^2 + P_n^2} + \paren {\paren {P_{n + 1}^2 + P_n^2 } - \paren {P_n^2 + P_{n - 1}^2 } }\) induction hypothesis
\(\ds \) \(=\) \(\ds 5 \paren {P_{n + 1}^2 + P_n^2} + \paren {P_{n + 1}^2 - P_{n - 1}^2 }\) simplifying
\(\ds \) \(=\) \(\ds 5 \paren {P_{n + 1}^2 + P_n^2} + \paren {P_{n + 1} + P_{n - 1} }\paren {P_{n + 1} - P_{n - 1} }\) Difference of Two Squares
\(\ds \) \(=\) \(\ds 5 P_{n + 1}^2 + 5 P_n^2 + \paren {P_{n + 1} + P_{n + 1} - 2 P_n} 2 P_n\) Definition of Pell Numbers
\(\ds \) \(=\) \(\ds 5 P_{n + 1}^2 + P_n^2 + 4 P_{n + 1} P_n\)
\(\ds \) \(=\) \(\ds P_{n + 1}^2 + P_n^2 + 4 P_n P_{n + 1} + 4 P_{n + 1}^2\)
\(\ds \) \(=\) \(\ds P_{n + 1}^2 + \paren {P_n + 2 P_{n + 1} }^2\) Square of Sum
\(\ds \) \(=\) \(\ds P_{n + 1}^2 + P_{n + 2}^2\) Definition of Pell Numbers


The result follows by induction.

$\blacksquare$

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