Pell number

In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2.

As well as being used to approximate the square root of two, Pell numbers can be used to find square triangular numbers, to construct integer approximations to the right isosceles triangle, and to solve certain combinatorial enumeration problems.

The Pell–Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type; the Pell and companion Pell numbers are Lucas sequences.

The first few terms of the sequence are Analogously to the Binet formula, the Pell numbers can also be expressed by the closed form formula

For large values of n, the (1 + √2)n term dominates this expression, so the Pell numbers are approximately proportional to powers of the silver ratio 1 + √2, analogous to the growth rate of Fibonacci numbers as powers of the golden ratio.

A third definition is possible, from the matrix formula Many identities can be derived or proven from these definitions; for instance an identity analogous to Cassini's identity for Fibonacci numbers, is an immediate consequence of the matrix formula (found by considering the determinants of the matrices on the left and right sides of the matrix formula).

[2] Pell numbers arise historically and most notably in the rational approximation to √2.

If two large integers x and y form a solution to the Pell equation then their ratio ⁠x/y⁠ provides a close approximation to √2.

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 BCE.

[3] The Greek mathematicians of the fifth century BCE also knew of this sequence of approximations:[4] Plato refers to the numerators as rational diameters.

[5] In the second century CE Theon of Smyrna used the term the side and diameter numbers to describe the denominators and numerators of this sequence.

[6] 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 √2 allows them to be used for accurate rational approximations to a regular octagon with vertex coordinates (± Pi, ± Pi +1) and (± Pi +1, ± Pi ).

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.

Falcón and Díaz-Barrero (2006) proved another identity relating Pell numbers to squares and showing that the sum of the Pell numbers up to P4n +1 is always a square: For instance, the sum of the Pell numbers up to P5, 0 + 1 + 2 + 5 + 12 + 29 = 49, is the square of P2 + P3 = 2 + 5 = 7.

As Martin (1875) describes, the Pell numbers can be used to form Pythagorean triples in which a and b are one unit apart, corresponding to right triangles that are nearly isosceles.

A near-isosceles Pythagorean triple is an integer solution to a 2 + b 2 = c 2 where a + 1 = b.

The next table shows that splitting the odd number Hn into nearly equal halves gives a square triangular number when n is even and a near isosceles Pythagorean triple when n is odd.

Since √2 is irrational, we cannot have ⁠H/P⁠ = √2, i.e., The best we can achieve is either The (non-negative) solutions to H  2 − 2P  2 = 1 are exactly the pairs (Hn, Pn) with n even, and the solutions to H  2 − 2P  2 = −1 are exactly the pairs (Hn, Pn) with n odd.

The equality c 2 = a 2 + (a + 1) 2 = 2a 2 + 2a + 1 occurs exactly when 2c 2 = 4a 2 + 4a + 2 which becomes 2P  2 = H  2 + 1 with the substitutions H = 2a + 1 and P = c. Hence the n-th solution is an = ⁠H2n +1 − 1/2⁠ and cn = P2n +1.