Perrin number

In mathematics, the Perrin numbers are defined by the recurrence relation

P (n) = P (n - 2) + P (n - 3)

for

n > 2

with initial values

P (0) = 3, P (1) = 0, P (2) = 2

The sequence of Perrin numbers starts with

3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39 ... (sequence A001608 in OEIS)

The number of different maximal independent sets in an $n$ -vertex cycle graph is counted by the $n$ th Perrin number for $n > 1$ .^[1]

History

This sequence was mentioned implicitly by Édouard Lucas (1876). In 1899, the same sequence was mentioned explicitly by François Olivier Raoul Perrin.^[2] The most extensive treatment of this sequence was given by Adams and Shanks (1982).

Properties

Generating function

The generating function of the Perrin sequence is

G(P(n);x)=\frac{3-x^2}{1-x^2-x^3}.

Matrix formula

\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}^n \begin{pmatrix} 3 \\ 0 \\ 2 \end{pmatrix} = \begin{pmatrix} P\left(n\right) \\ P\left(n+1\right) \\ P\left(n+2\right) \end{pmatrix}

Binet-like formula

The Perrin sequence numbers can be written in terms of powers of the roots of the equation

x^3 -x -1 = 0.

This equation has 3 roots; one real root p (known as the plastic number) and two complex conjugate roots q and r. Given these three roots, the Perrin sequence analogue of the Lucas sequence Binet formula is

P\left(n\right) = {p^n} + {q^n} + {r^n}.

Since the magnitudes of the complex roots q and r are both less than 1, the powers of these roots approach 0 for large n. For large n the formula reduces to

P\left(n\right) \approx {p^n}

This formula can be used to quickly calculate values of the Perrin sequence for large n. The ratio of successive terms in the Perrin sequence approaches p, a.k.a. the plastic number, which has a value of approximately 1.324718. This constant bears the same relationship to the Perrin sequence as the golden ratio does to the Lucas sequence. Similar connections exist also between p and the Padovan sequence, between the golden ratio and Fibonacci numbers, and between the silver ratio and Pell numbers.

Multiplication formula

From the Binet formula, we can obtain a formula for G(kn) in terms of G(n−1), G(n) and G(n+1); we know

\begin{matrix} G(n-1) & = &p^{-1}p^n + &q^{-1}q^n +& r^{-1} r^n\\ G(n) & =& p^n+&q^n+&r^n\\ G(n+1) &=& pp^n +& qq^n +& rr^n\end{matrix}

which gives us three linear equations with coefficients over the splitting field of $x^3 -x -1$ ; by inverting a matrix we can solve for $p^n, q^n, r^n$ and then we can raise them to the kth power and compute the sum.

Example magma code:

P<x> := PolynomialRing(Rationals());
S<t> := SplittingField(x^3-x-1);
P2<y> := PolynomialRing(S);
p,q,r := Explode([r[1] : r in Roots(y^3-y-1)]);
Mi:=Matrix([[1/p,1/q,1/r],[1,1,1],[p,q,r]])^(-1);
T<u,v,w> := PolynomialRing(S,3);
v1 := ChangeRing(Mi,T) *Matrix([[u],[v],[w]]);
[p^i*v1[1,1]^3 + q^i*v1[2,1]^3 + r^i*v1[3,1]^3 : i in [-1..1]];

with the result that, if we have $u = G(n-1), v = G(n), w = G(n+1)$ , then

\begin{matrix} 23G(2n-1) &=& 4u^2 + 3v^2 + 9w^2 + 18uv - 12uw - 4vw \\ 23G(2n) &=& - 6u^2 + 7v^2 - 2w^2 - 4uv + 18uw + 6vw\\ 23G(2n+1) &=& 9u^2 + v^2 + 3w^2 + 6uv - 4uw + 14vw \\ 23G(3n-1)& = &\left(-4u^3 + 2v^3 -w^3 + 9(uv^2+vw^2+wu^2) + 3v^2w+6uvw\right)\\ 23G(3n)& = &\left(3u^3 + 2v^3 + 3w^3 - 3(uv^2 + uw^2 + vw^2 + vu^2) + 6v^2w + 18uvw\right) \\ 23G(3n+1)& = &\left(v^3-w^3+6uv^2+9uw^2+6vw^2+9vu^2-3wu^2+6wv^2-6uvw\right) \end{matrix}

The number 23 here arises from the discriminant of the defining polynomial of the sequence.

This allows you to compute the nth Perrin number using integer arithmetic in $O(\log n)$ multiplies.

Primes and divisibility

Perrin pseudoprimes

It has been proven that for all primes p, p divides P(p). However, the converse is not true: for some composite numbers n, n may still divide P(n). If n has this property, it is called a Perrin pseudoprime.

The first few Perrin pseudoprimes are

271441, 904631, 16532714, 24658561, 27422714, 27664033, 46672291, 102690901, 130944133, 196075949, 214038533, 517697641, 545670533, 801123451, 855073301, 903136901, 970355431, ... (sequence A013998 in OEIS)

The question of the existence of Perrin pseudoprimes was considered by Perrin himself, but it was not known whether they existed until Adams and Shanks (1982) discovered the smallest one, 271441 = 521²; the next-smallest is 904631 = 7 x 13 x 9941. There are seventeen of them less than a billion;^[3] Jon Grantham has proved^[4] that there are infinitely many Perrin pseudoprimes.

Perrin primes

A Perrin prime is a Perrin number that is prime. The first few Perrin primes are:

2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797, ... (sequence A074788 in OEIS)

For these Perrin primes, the index $n$ of $P (n)$ is

2, 3, 4, 5, 6, 7, 10, 12, 20, 21, 24, 34, 38, 75, 122, 166, 236, 355, 356, 930, 1042, 1214, 1461, 1622, 4430, 5802, 9092, ... (sequence A112881 in OEIS)

E. W. Weisstein found a 32,147 digit probable Perrin prime $P (263226)$ in May 2006.

Notes

↑ Füredi (1987)
↑ Knuth (2011)
↑ (sequence A013998 in OEIS)
↑ Jon Grantham (2010). "There are infinitely many Perrin pseudoprimes" (PDF). Journal of Number Theory 130 (5): 1117–1128. doi:10.1016/j.jnt.2009.11.008.

References

Adams, William; Shanks, Daniel (1982). "Strong primality tests that are not sufficient". Mathematics of Computation (American Mathematical Society) 39 (159): 255–300. doi:10.2307/2007637. JSTOR 2007637. MR 0658231.

Füredi, Z. (1987). "The number of maximal independent sets in connected graphs". Journal of Graph Theory 11 (4): 463–470. doi:10.1002/jgt.3190110403.

Knuth, Donald E. (2011). The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1. Addison-Wesley. ISBN 0201038048.

Lucas, E. (1878). "Théorie des fonctions numériques simplement périodiques". American Journal of Mathematics (The Johns Hopkins University Press) 1 (3): 197–240. doi:10.2307/2369311. JSTOR 2369311.

Perrin, R. (1899). "Query 1484". L'Intermédiaire des Mathématiciens 6: 76.

External links

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