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.
Read more about this topic: Pell Number
Famous quotes containing the word squares:
“And New York is the most beautiful city in the world? It is not far from it. No urban night is like the night there.... Squares after squares of flame, set up and cut into the aether. Here is our poetry, for we have pulled down the stars to our will.”
—Ezra Pound (1885–1972)