Pell number

The sides of the squares used to construct a silver spiral are the Pell numbers

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. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell-Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82.

Both the Pell numbers and the companion Pell numbers may be calculated by means of a recurrence relation similar to that for the Fibonacci numbers, and both sequences of numbers grow exponentially, proportionally to powers of the silver ratio 1 + √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.^[1]

As with Pell's equation, the name of the Pell numbers stems from Leonhard Euler's mistaken attribution of the equation and the numbers derived from it to John Pell. 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.

Pell numbers

The Pell numbers are defined by the recurrence relation

$P_n=\begin{cases}0&\mbox{if }n=0;\\1&\mbox{if }n=1;\\2P_{n-1}+P_{n-2}&\mbox{otherwise.}\end{cases}$

In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are

0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, ... (sequence A000129 in OEIS).

The Pell numbers can also be expressed by the closed form formula

$P_n=\frac{(1+\sqrt2)^n-(1-\sqrt2)^n}{2\sqrt2}.$

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

A third definition is possible, from the matrix formula

$\begin{pmatrix} P_{n+1} & P_n \\ P_n & P_{n-1} \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 0 \end{pmatrix}^n.$

Many identities can be derived or proven from these definitions; for instance an identity analogous to Cassini's identity for Fibonacci numbers,

$P_{n+1}P_{n-1}-P_n^2 = (-1)^n,$

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]

Approximation to the square root of two

Rational approximations to regular octagons, with coordinates derived from the Pell numbers.

Pell numbers arise historically and most notably in the rational approximation to the square root of 2. If two large integers x and y form a solution to the Pell equation

$\displaystyle x^2-2y^2=\pm 1,$

then their ratio $\tfrac{x}{y}$ provides a close approximation to $\scriptstyle\sqrt 2$. The sequence of approximations of this form is

$1, \frac32, \frac75, \frac{17}{12}, \frac{41}{29}, \frac{99}{70}, \dots$

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 $\tfrac{P_{n-1}+P_n}{P_n}$. The approximation

$\sqrt 2\approx\frac{577}{408}$

of this type was known to Indian mathematicians in the third or fourth century B.C.^[3] The Greek mathematicians of the fifth century B.C. also knew of this sequence of approximations:^[4] Plato refers to the numerators as rational diameters.^[5] 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.^[6]

These approximations can be derived from the continued fraction expansion of $\scriptstyle\sqrt 2$:

$\sqrt 2 = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots\,}}}}}.$

Truncating this expansion to any number of terms produces one of the Pell-number-based approximations in this sequence; for instance,

$\frac{577}{408} = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2}}}}}}}.$

As Knuth (1994) describes, the fact that Pell numbers approximate $\scriptstyle\sqrt 2$ allows them to be used for accurate rational approximations to a regular octagon with vertex coordinates $(\pm P_i,\pm P_{i+1})$ and $(\pm P_{i+1},\pm P_i)$. All vertices are equally distant from the origin, and form nearly uniform angles around the origin. Alternatively, the points $(\pm(P_i+P_{i-1}),0)$, $(0,\pm(P_i+P_{i-1}))$, and $(\pm P_i,\pm P_i)$ 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, ... (sequence A086383 in OEIS).

For these ns are

2, 3, 5, 11, 13, 29, 41, 53, 59, 89, 97, 101, 167, 181, 191, ... (sequence A096650 in OEIS)

As with the Fibonacci numbers, a Pell number $P_n$ can only be prime if n itself is prime, because a divides b if and only if $P_a$ divides $P_b$.

If and only if a prime p congruent to 1 or 7 (mod 8), then p divides P_p-1, otherwise, p divides P_p+1. (The only exception is p = 2, if and only if p = 2, then p divides P_p)

The only Pell numbers that are squares, cubes, or any higher power of an integer are 0, 1, and 169 = 13².^[7]

However, despite having so few squares or other powers, Pell numbers have a close connection to square triangular numbers.^[8] Specifically, these numbers arise from the following identity of Pell numbers:

$\bigl((P_{k-1}+P_k)\cdot P_k\bigr)^2 = \frac{(P_{k-1}+P_k)^2\cdot\left((P_{k-1}+P_k)^2-(-1)^k\right)}{2}.$

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 $P_{4n+1}$ is always a square:

$\sum_{i=0}^{4n+1} P_i = \left(\sum_{r=0}^n 2^r{2n+1\choose 2r}\right)^2 = (P_{2n}+P_{2n+1})^2.$

For instance, the sum of the Pell numbers up to $P_5$, $0+1+2+5+12+29=49$, is the square of $P_2+P_3=2+5=7$. The numbers $P_{2n}+P_{2n+1}$ 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.

Pythagorean triples

Integer right triangles with nearly equal legs, derived from the Pell numbers.

If a right triangle has integer side lengths a, b, c (necessarily satisfying the Pythagorean theorem a²+b²=c²), then (a,b,c) is known as a Pythagorean triple. 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. Each such triple has the form

$(2P_{n}P_{n+1}, P_{n+1}^2 - P_{n}^2, P_{n+1}^2 + P_{n}^2=P_{2n+1}).$

The sequence of Pythagorean triples formed in this way is

(4,3,5), (20,21,29), (120,119,169), (696,697,985), ....

Pell-Lucas numbers

The companion Pell numbers or Pell-Lucas numbers are defined by the recurrence relation

$Q_n=\begin{cases}2&\mbox{if }n=0;\\2&\mbox{if }n=1;\\2Q_{n-1}+Q_{n-2}&\mbox{otherwise.}\end{cases}$

In words: the first two numbers in the sequence are both 2, and each successive number is formed by adding twice the previous Pell-Lucas number to the Pell-Lucas number before that, or equivalently, by adding the next Pell number to the previous Pell number: thus, 82 is the companion to 29, and 82 = 2 * 34 + 14 = 70 + 12. The first few terms of the sequence are (sequence A002203 in OEIS): 2, 2, 6, 14, 34, 82, 198, 478...

Like the Fibonacci number to Lucas number, $Q_n=\frac{P_{2n}}{P_n}$ for all natural number n.

The companion Pell numbers can be expressed by the closed form formula

$Q_n=(1+\sqrt 2)^n+(1-\sqrt 2)^n.$

These numbers are all even; each such number is twice the numerator in one of the rational approximations to $\scriptstyle\sqrt 2$ discussed above.

Like the Lucas sequence, if a Pell-Lucas number $\frac{Q_n}{2}$ is prime, it is necessary that n be either prime or a power of 2. The Pell-Lucas primes are

3, 7, 17, 41, 239, 577, ... (sequence A086395 in OEIS).

For these ns are

2, 3, 4, 5, 7, 8, 16, 19, 29, 47, 59, 163, 257, 421, ... (sequence A099088 in OEIS).

Computations and connections

The following table gives the first few powers of the silver ratio $\delta=\delta_S=1+\sqrt2$ and its conjugate $\bar{\delta}=1-\sqrt{2}.$

$n$	$(1+\sqrt{2})^n$	$(1-\sqrt{2})^n$
0	$1+0\sqrt{2}=1.0$	$1-0\sqrt{2}=1.0$
1	$1+1\sqrt{2}=2.41421\ldots$	$1-1\sqrt{2}=-0.41421\ldots$
2	$3+2\sqrt{2}=5.82842\ldots$	$3-2\sqrt{2}=0.17157\ldots$
3	$7+5\sqrt{2}=14.07106\ldots$	$7-5\sqrt{2}=-0.07106\ldots$
4	$17+12\sqrt{2}=33.97056\ldots$	$17-12\sqrt{2}=0.02943\ldots$
5	$41+29\sqrt{2}=82.01219\ldots$	$41-29\sqrt{2}=-0.01219\ldots$
6	$99+70\sqrt{2}=197.9949\ldots$	$99-70\sqrt{2}=0.0050\ldots$
7	$239+169\sqrt{2}=478.00209\ldots$	$239-169\sqrt{2}=-0.00209\ldots$
8	$577+408\sqrt{2}=1153.99913\ldots$	$577-408\sqrt{2}=0.00086\ldots$
9	$1393+985\sqrt{2}=2786.00035\ldots$	$1393-985\sqrt{2}=-0.00035\ldots$
10	$3363+2378\sqrt{2}=6725.99985\ldots$	$3363-2378\sqrt{2}=0.00014\ldots$
11	$8119+5741\sqrt{2}=16238.00006\ldots$	$8119-5741\sqrt{2}=-0.00006\ldots$
12	$19601+13860\sqrt{2}=39201.99997\ldots$	$19601-13860\sqrt{2}=0.00002\ldots$

The coefficients are the half companion Pell numbers $H_n$ and the Pell numbers $P_n$ which are the (non-negative) solutions to $H^2-2P^2=\pm1.$ A square triangular number is a number $\textstyle N=\frac{t(t+1)}{2}=s^2\,$, which is both the tth triangular number and the sth square number. 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 $H_n$ into nearly equal halves gives a square triangular number when n is even and a near isosceles Pythagorean triple when n is odd. All solutions arise in this manner.

$n$	$H_n$	$P_n$	t	t+1	s	a	b	c
0	1	0	0	0	0
1	1	1				0	1	1
2	3	2	1	2	1
3	7	5				3	4	5
4	17	12	8	9	6
5	41	29				20	21	29
6	99	70	49	50	35
7	239	169				119	120	169
8	577	408	288	289	204
9	1393	985				696	697	985
10	3363	2378	1681	1682	1189
11	8119	5741				4059	4060	5741
12	19601	13860	9800	9801	6930

Definitions

The half companion Pell Numbers $H_n$ and the Pell numbers $P_n$ can be derived in a number of easily equivalent ways.

Raising to powers:

$(1+\sqrt2)^n=H_n+P_n\sqrt{2}$

$(1-\sqrt2)^n=H_n-P_n\sqrt{2}.$

From this it follows that there are closed forms:

$H_n=\frac{(1+\sqrt2)^n+(1-\sqrt2)^n}{2}.$

and

$P_n\sqrt2=\frac{(1+\sqrt2)^n-(1-\sqrt2)^n}{2}.$

Paired recurrences:

$H_n=\begin{cases}1&\mbox{if }n=0;\\H_{n-1}+2P_{n-1}&\mbox{otherwise.}\end{cases}$

$P_n=\begin{cases}0&\mbox{if }n=0;\\H_{n-1}+P_{n-1}&\mbox{otherwise.}\end{cases}$

and matrix formulations:

$\begin{pmatrix} H_n \\ P_n \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} H_{n-1} \\ P_{n-1} \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 1 & 1 \end{pmatrix}^n \begin{pmatrix} 1 \\ 0 \end{pmatrix}.$

So

$\begin{pmatrix} H_n & 2P_n \\ P_n & H_n \end{pmatrix}= \begin{pmatrix} 1 & 2 \\ 1 & 1 \end{pmatrix}^n .$

Approximations

The difference between $H_n\,$ and $P_n\sqrt2\,$ is $(1-\sqrt2)^n \approx (-0.41421)^n,$ which goes rapidly to zero. So $(1+\sqrt2)^n=H_n+P_n\sqrt2\,$ is extremely close to $2H_n.$

From this last observation it follows that the integer ratios $H_n/P_n\,$ rapidly approach $\sqrt2,$ and $H_n / H_{n-1}\,$ and $P_n/P_{n-1}\,$ rapidly approach $1+\sqrt2.$

H² − 2P² = ±1

Since $\sqrt2$ is irrational, we can't have $H/P=\sqrt2,$ i.e., $H^2/P^2=2P^2/P^2.$ The best we can achieve is either $H^2/P^2=(2P^2-1)/P^2\,$ or $H^2/P^2=(2P^2+1)/P^2.$

The (non-negative) solutions to $H^2-2P^2=1 \,$ are exactly the pairs $H_n,P_n \mbox{ with } n\,$ even and the solutions to $H^2-2P^2=-1 \,$ are exactly the pairs $H_n,P_n \mbox{ with } n \,$ odd. To see this, note first that

$H_{n+1}^2-2P_{n+1}^2=(H_n+2P_n)^2- 2(H_n+P_n)^2=-(H_n^2-2P_n^2),$

so that these differences, starting with $H_{0}^2-2P_{0}^2=1,$ are alternately $1 \mbox{ and }-1.$ Then note that every positive solution comes in this way from a solution with smaller integers since $(2P-H)^2-2(H-P)^2=-(H^2-2P^2).$ The smaller solution also has positive integers with the one exception $H=P=1,$ which comes from $H_0=1 \mbox{ and }P_0=0.$

Square triangular numbers

The required equation $t(t+1)/2=s^2\,$ is equivalent to $4t^2+4t+1=8s^2+1,$ which becomes $H^2=2P^2+1$ with the substitutions $H=2t+1 \mbox{ and } P=2s.\,$ Hence the nth solution is $t_n=(H_{2n}-1)/2$ and $s_n=P_{2n}/2.$

Observe that $t$ and $t+1$ are relatively prime so that $t(t+1)/2=s^2\,$ happens exactly when they are adjacent integers, one a square $H^2\,$ and the other twice a square $2P^2.$ Since we know all solutions of that equation, we also have

$t_n=\begin{cases}2P_n^2&\mbox{if }n\mbox{ is even};\\H_{n}^2&\mbox{if }n\mbox{ is odd.}\end{cases}$

and $s_n=H_nP_n.$

This alternate expression is seen in the next table.

$n$	$H_n$	$P_n$		t	t+1	s		a	b	c
0	1	0
1	1	1		1	2	1		3	4	5
2	3	2		8	9	6		20	21	29
3	7	5		49	50	35		119	120	169
4	17	12		288	289	204		696	697	985
5	41	29		1681	1682	1189		4059	4060	5741
6	99	70		9800	9801	6930		23660	23661	33461

Pythagorean triples

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 $H=2a+1 \mbox{ and } P=c.\,$ Hence the nth solution is $a_n=(H_{2n+1}-1)/2$ and $c_n={P_{2n+1}}.$

The table above shows that, in one order or the other, $a_n\mbox{ and }b_n=a_n+1$ are $H_nH_{n+1}\mbox{ and }2P_nP_{n+1}$ while $c_n=H_{n+1}P_n+P_{n+1}H_n.$

Notes

1. For instance, Sellers (2002) proves that the number of perfect matchings in the Cartesian product of a path graph and the graph K₄-e can be calculated as the product of a Pell number with the corresponding Fibonacci number.
2. For the matrix formula and its consequences see Ercolano (1979) and Kilic and Tasci (2005). Additional identities for the Pell numbers are listed by Horadam (1971) and Bicknell (1975).
3. As recorded in the Shulba Sutras; see e.g. Dutka (1986), who cites Thibaut (1875) for this information.
4. See Knorr (1976) for the fifth century date, which matches Proclus' claim that the side and diameter numbers were discovered by the Pythagoreans. For more detailed exploration of later Greek knowledge of these numbers see Thompson (1929), Vedova (1951), Ridenhour (1986), Knorr (1998), and Filep (1999).
5. For instance, as several of the references from the previous note observe, in Plato's Republic there is a reference to the "rational diameter of 5", by which Plato means 7, the numerator of the approximation 7/5 of which 5 is the denominator.
6. Heath, Sir Thomas Little (1921), History of Greek Mathematics: From Thales to Euclid, Courier Dover Publications, p. 112, ISBN 9780486240732.
7. 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.)
8. Sesskin (1962). See the square triangular number article for a more detailed derivation.

References

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• Pethő, A. (1992). "The Pell sequence contains only trivial perfect powers". Sets, graphs, and numbers (Budapest, 1991). Colloq. Math. Soc. János Bolyai, 60, North-Holland. pp. 561–568. MR 1218218.
• Ridenhour, J. R. (1986). "Ladder approximations of irrational numbers". Mathematics Magazine 59 (2): 95–105. doi:10.2307/2690427. JSTOR 2690427.
• Santana, S. F.; Diaz-Barrero, J. L. (2006). "Some properties of sums involving Pell numbers" (PDF). Missouri Journal of Mathematical Sciences 18 (1).
• Sellers, James A. (2002). "Domino tilings and products of Fibonacci and Pell numbers" (PDF). Journal of Integer Sequences 5. MR 1919941.
• Sesskin, Sam (1962). "A "converse" to Fermat's last theorem?". Mathematics Magazine 35 (4): 215–217. doi:10.2307/2688551. JSTOR 2688551.
• Thibaut, George (1875). "On the Súlvasútras". Journal of the Royal Asiatic Society of Bengal 44: 227–275.
• Thompson, D'Arcy Wentworth (1929). "III.—Excess and defect: or the little more and the little less". Mind: New Series 38 (149): 43–55. JSTOR 2249223.
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External links

• Weisstein, Eric W., "Pell Number", MathWorld.