of two Pell numbers arise historically and most notably in the rational approximation to . If two large integers x and y form a solution to the Pell equation then their ratio provides a close approximation to . The sequence of approximations of this form is where the denominator of each fraction is a Pell number and the numerator is the sum of a Pell number and its predecessor in the sequence. That is, the solutions have the form The approximation of this type was known to Indian mathematicians in the third or fourth century B.C. The Greek mathematicians of the fifth century B.C. also knew of this sequence of approximations: Plato refers to the numerators as rational diameters. In the 2nd century CE Theon of Smyrna used the term the side and diameter numbers to describe the denominators and numerators of this sequence. These approximations can be derived from the continued fraction expansion of : Truncating this expansion to any number of terms produces one of the Pell-number-based approximations in this sequence; for instance, As Knuth (1994) describes, the fact that Pell numbers approximate allows them to be used for accurate rational approximations to a regular octagon with vertex coordinates and . All vertices are equally distant from the origin, and form nearly uniform angles around the origin. Alternatively, the points , , and form approximate octagons in which the vertices are nearly equally distant from the origin and form uniform angles. Primes and squares A Pell prime is a Pell number that is prime. The first few Pell primes are 2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, ... . 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, ... These indices are all themselves prime. As with the Fibonacci numbers, a Pell number Pn can only be prime if n itself is prime, because if d is a divisor of n then Pd is a divisor of Pn. 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