Editing Pell number (section)

== Primes and squares ==

A '''Pell prime''' is a Pell number that is [[prime number|prime]]. The first few Pell primes are
:2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, ... {{OEIS|id=A086383}}.
The indices of these primes within the sequence of all Pell numbers are
:2, 3, 5, 11, 13, 29, 41, 53, 59, 89, 97, 101, 167, 181, 191, 523, 929, 1217, 1301, 1361, 2087, 2273, 2393, 8093, ... {{OEIS|id=A096650}}
These indices are all themselves prime. As with the Fibonacci numbers, a Pell number ''P''<sub>''n''</sub> can only be prime if ''n'' itself is prime, because if ''d'' is a [[divisor]] of ''n'' then ''P''<sub>''d''</sub> is a divisor of ''P''<sub>''n''</sub>.

The only Pell numbers that are [[square number|squares]], [[perfect cube|cubes]], or any higher [[perfect power|power of an integer]] are 0, 1, and 169 = 13<sup>2</sup>.<ref>Pethő (1992); Cohn (1996). Although the Fibonacci numbers are defined by a very similar recurrence to the Pell numbers, Cohn writes that an analogous result for the Fibonacci numbers seems much more difficult to prove. (However, this was proven in 2006 by Bugeaud et al.)</ref>

However, despite having so few squares or other powers, Pell numbers have a close connection to [[square triangular number]]s.<ref>Sesskin (1962). See the [[square triangular number]] article for a more detailed derivation.</ref> Specifically, these numbers arise from the following identity of Pell numbers:
:<math>\bigl(\left(P_{k-1}+P_k\right)\cdot P_k\bigr)^2 = \frac{\left(P_{k-1}+P_k\right)^2\cdot\left(\left(P_{k-1}+P_k\right)^2-(-1)^k\right)}{2}.</math>
The left side of this identity describes a square number, while the right side describes a [[triangular number]], so the result is a square triangular number.

Falcón and Díaz-Barrero (2006) proved another identity relating Pell numbers to squares and showing that the sum of the Pell numbers up to ''P''<sub>4''n''+1</sub> is always a square:
:<math>\sum_{i=0}^{4n+1} P_i = \left(\sum_{r=0}^n 2^r{2n+1\choose 2r}\right)^{\!2} = \left(P_{2n}+P_{2n+1}\right)^2.</math>
For instance, the sum of the Pell numbers up to ''P''<sub>5</sub>, {{nowrap|1=0 + 1 + 2 + 5 + 12 + 29 = 49}}, is the square of {{nowrap|1=''P''<sub>2</sub> + ''P''<sub>3</sub> = 2 + 5 = 7}}. The numbers {{nowrap|''P''<sub>2''n''</sub> + ''P''<sub>2''n''+1</sub>}} forming the square roots of these sums,
:1, 7, 41, 239, 1393, 8119, 47321, … {{OEIS|id=A002315}},
are known as the [[Newman–Shanks–Williams number|Newman–Shanks–Williams (NSW) numbers]].