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. 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.
Contents
Pell numbers
The Pell numbers are defined by the recurrence relation
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
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]}
Approximation to the square root of two
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 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 :
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 (±P_{i}, ±P_{i+1}) and (±P_{i+1}, ±P_{i}). 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
The indices of these primes within the sequence of all Pell numbers are
These indices are all themselves prime. As with the Fibonacci numbers, a Pell number P_{n} can only be prime if n itself is prime, because if d is a divisor of n then P_{d} is a divisor of P_{n}.
The only Pell numbers that are squares, cubes, or any higher power of an integer are 0, 1, and 169 = 13^{2}.^{[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:
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) proved 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:
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,
are known as the Newman–Shanks–Williams (NSW) numbers.
Pythagorean triples
If a right triangle has integer side lengths a, b, c (necessarily satisfying the Pythagorean theorem a^{2} + b^{2} = c^{2}), 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
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
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 the OEIS): 2, 2, 6, 14, 34, 82, 198, 478,…
Like the relationship between Fibonacci numbers and Lucas numbers,
for all natural numbers n.
The companion Pell numbers can be expressed by the closed form formula
These numbers are all even; each such number is twice the numerator in one of the rational approximations to discussed above.
Like the Lucas sequence, if a Pell–Lucas number 1/2Q_{n} is prime, it is necessary that n be either prime or a power of 2. The Pell–Lucas primes are
For these n are
Computations and connections
The following table gives the first few powers of the silver ratio δ = δ_{S} = 1 + √2 and its conjugate δ = 1 − √2.
n (1 + √2)^{n} (1 − √2)^{n} 0 1 + 0√2 = 1 1 − 0√2 = 1 1 1 + 1√2 = 2.41421… 1 − 1√2 = −0.41421… 2 3 + 2√2 = 5.82842… 3 − 2√2 = 0.17157… 3 7 + 5√2 = 14.07106… 7 − 5√2 = −0.07106… 4 17 + 12√2 = 33.97056… 17 − 12√2 = 0.02943… 5 41 + 29√2 = 82.01219… 41 − 29√2 = −0.01219… 6 99 + 70√2 = 197.9949… 99 − 70√2 = 0.0050… 7 239 + 169√2 = 478.00209… 239 − 169√2 = −0.00209… 8 577 + 408√2 = 1153.99913… 577 − 408√2 = 0.00086… 9 1393 + 985√2 = 2786.00035… 1393 − 985√2 = −0.00035… 10 3363 + 2378√2 = 6725.99985… 3363 − 2378√2 = 0.00014… 11 8119 + 5741√2 = 16238.00006… 8119 − 5741√2 = −0.00006… 12 19601 + 13860√2 = 39201.99997… 19601 − 13860√2 = 0.00002…
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} = ±1. A square triangular number is a number
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
From this it follows that there are closed forms:
and
Paired recurrences
Matrix formulations
So
Approximations
The difference between H_{n} and P_{n}√2 is
which goes rapidly to zero. So
is extremely close to 2H_{n}.
From this last observation it follows that the integer ratios H_{n}/P_{n} rapidly approach √2; and H_{n}/H_{n−1} and P_{n}/P_{n−1} rapidly approach 1 + √2.
H^{2} − 2P^{2} = ±1
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 (H_{n}, P_{n}) with n even, and the solutions to H^{2} − 2P^{2} = −1 are exactly the pairs (H_{n}, P_{n}) with n odd. To see this, note first that
so that these differences, starting with H^{2}
_{0} − 2P^{2}
_{0} = 1, are alternately
1 and −1. Then note that every positive solution comes in this way from a solution with smaller integers since
The smaller solution also has positive integers, with the one exception: H = P = 1 which comes from H_{0} = 1 and P_{0} = 0.
Square triangular numbers
The required equation
is equivalent to: which becomes H^{2} = 2P^{2} + 1 with the substitutions H = 2t + 1 and P = 2s. Hence the nth solution is
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
and
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 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} and b_{n} = a_{n} + 1 are H_{n}H_{n+1} and 2P_{n}P_{n+1} while c_{n} = H_{n+1}P_{n} + P_{n+1}H_{n}.
Notes
- ^ For instance, Sellers (2002) proves that the number of perfect matchings in the Cartesian product of a path graph and the graph K_{4} − e can be calculated as the product of a Pell number with the corresponding Fibonacci number.
- ^ 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).
- ^ As recorded in the Shulba Sutras; see e.g. Dutka (1986), who cites Thibaut (1875) for this information.
- ^ 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).
- ^ 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.
- ^ Heath, Sir Thomas Little (1921), History of Greek Mathematics: From Thales to Euclid, Courier Dover Publications, p. 112, ISBN 9780486240732.
- ^ 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.)
- ^ Sesskin (1962). See the square triangular number article for a more detailed derivation.
References
- Bicknell, Marjorie (1975). "A primer on the Pell sequence and related sequences". Fibonacci Quarterly. 13 (4): 345–349. MR 0387173.
- Cohn, J. H. E. (1996). "Perfect Pell powers". Glasgow Mathematical Journal. 38 (1): 19–20. doi:10.1017/S0017089500031207. MR 1373953.
- Dutka, Jacques (1986). "On square roots and their representations". Archive for History of Exact Sciences. 36 (1): 21–39. doi:10.1007/BF00357439. MR 0863340.
- Ercolano, Joseph (1979). "Matrix generators of Pell sequences". Fibonacci Quarterly. 17 (1): 71–77. MR 0525602.
- Filep, László (1999). "Pythagorean side and diagonal numbers" (PDF). Acta Mathematica Academiae Paedagogiace Nyíregyháziensis. 15: 1–7.
- Horadam, A. F. (1971). "Pell identities". Fibonacci Quarterly. 9 (3): 245–252, 263. MR 0308029.
- Kilic, Emrah; Tasci, Dursun (2005). "The linear algebra of the Pell matrix". Boletín de la Sociedad Matemática Mexicana, Tercera Serie. 11 (2): 163–174. MR 2207722.
- Knorr, Wilbur (1976). "Archimedes and the measurement of the circle: A new interpretation". Archive for History of Exact Sciences. 15 (2): 115–140. doi:10.1007/BF00348496. MR 0497462.
- Knorr, Wilbur (1998). ""Rational diameters" and the discovery of incommensurability". American Mathematical Monthly. 105 (5): 421–429. doi:10.2307/3109803. JSTOR 3109803.
- Knuth, Donald E. (1994). "Leaper graphs". The Mathematical Gazette. 78 (483): 274–297. arXiv:math.CO/9411240 . doi:10.2307/3620202. JSTOR 3620202.
- Martin, Artemas (1875). "Rational right angled triangles nearly isosceles". The Analyst. 3 (2): 47–50. doi:10.2307/2635906. JSTOR 2635906.
- 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). Archived from the original (PDF) on 2007-05-08.
- 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.
- Vedova, G. C. (1951). "Notes on Theon of Smyrna". American Mathematical Monthly. 58 (10): 675–683. doi:10.2307/2307978. JSTOR 2307978.
External links
- Weisstein, Eric W. "Pell Number". MathWorld.
- OEIS sequence A001333 (Numerators of continued fraction convergents to sqrt(2)) -- The numerators of the same sequence of approximations