Pell Number - Primes and Squares

Primes and Squares

A Pell prime is a Pell number that is prime. The first few Pell primes are

2, 5, 29, 5741, ... (sequence A086383 in OEIS).

As with the Fibonacci numbers, a Pell number can only be prime if n itself is prime.

The only Pell numbers that are squares, cubes, or any higher power of an integer are 0, 1, and 169 = 132.

However, despite having so few squares or other powers, Pell numbers have a close connection to square triangular numbers. Specifically, these numbers arise from the following identity of Pell numbers:

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.

Santana and Diaz-Barrero (2006) prove another identity relating Pell numbers to squares and showing that the sum of the Pell numbers up to is always a square:

For instance, the sum of the Pell numbers up to, is the square of . The numbers forming the square roots of these sums,

1, 7, 41, 239, 1393, 8119, 47321, ... (sequence A002315 in OEIS),

are known as the Newman–Shanks–Williams (NSW) numbers.