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| In [[mathematics]], the '''Perrin numbers''' are defined by the [[recurrence relation]]
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| :''P''(0) = 3, ''P''(1) = 0, ''P''(2) = 2,
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| :''P''(''n'') = ''P''(''n'' − 2) + ''P''(''n'' − 3) for ''n'' > 2.
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| The sequence of Perrin numbers starts with
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| :[[3 (number)|3]], [[0 (number)|0]], [[2 (number)|2]], 3, 2, [[5 (number)|5]], 5, [[7 (number)|7]], [[10 (number)|10]], [[12 (number)|12]], [[17 (number)|17]], [[22 (number)|22]], [[29 (number)|29]], [[39 (number)|39]] ... {{OEIS|id=A001608}}
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| The number of different [[maximal independent set]]s in an ''n''-vertex [[cycle graph]] is counted by the ''n''th Perrin number for ''n'' > 1.<ref>{{harvtxt|Füredi|1987}}</ref>
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| ==History==
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| This sequence was mentioned implicitly by [[Édouard Lucas]] (1876). In 1899, the same sequence was mentioned explicitly by
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| François Olivier Raoul Perrin.<ref>{{harvtxt|Knuth|2011}}</ref> The most extensive treatment of this sequence was given by Adams and Shanks (1982).
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| == Properties ==
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| ===Generating function===
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| The [[generating function]] of the Perrin sequence is
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| :<math>G(P(n);x)=\frac{3-x^2}{1-x^2-x^3}.</math>
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| ===Matrix formula===
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| :<math> \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}^n
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| \begin{pmatrix} 3 \\ 0 \\ 2 \end{pmatrix} =
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| \begin{pmatrix} P\left(n\right) \\ P\left(n+1\right) \\ P\left(n+2\right) \end{pmatrix}
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| </math>
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| ===Binet-like formula===
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| The Perrin sequence numbers can be written in terms of powers of the roots of the equation
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| :<math> x^3 -x -1 = 0.</math>
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| 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
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| :<math>P\left(n\right) = {p^n} + {q^n} + {r^n}. </math>
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| 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
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| :<math>P\left(n\right) \approx {p^n} </math>
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| 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 number|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 number]]s.
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| ===Multiplication formula===
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| 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
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| :<math>
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| \begin{matrix}
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| G(n-1) & = &p^{-1}p^n + &q^{-1}q^n +& r^{-1} r^n\\
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| G(n) & =& p^n+&q^n+&r^n\\
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| G(n+1) &=& pp^n +& qq^n +& rr^n\end{matrix}</math>
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| which gives us three linear equations with coefficients over the [[splitting field]] of <math> x^3 -x -1 </math>; by inverting a matrix we can solve for <math>p^n, q^n, r^n</math> and then we can raise them to the ''k''th power and compute the sum.
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| Example [[Magma computer algebra system|magma]] code:
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| P<x> := PolynomialRing(Rationals());
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| S<t> := SplittingField(x^3-x-1);
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| P2<y> := PolynomialRing(S);
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| p,q,r := Explode([r[1] : r in Roots(y^3-y-1)]);
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| Mi:=Matrix([[1/p,1/q,1/r],[1,1,1],[p,q,r]])^(-1);
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| T<u,v,w> := PolynomialRing(S,3);
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| v1 := ChangeRing(Mi,T) *Matrix([[u],[v],[w]]);
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| [p^i*v1[1,1]^3 + q^i*v1[2,1]^3 + r^i*v1[3,1]^3 : i in [-1..1]];
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| with the result that, if we have <math>u = G(n-1), v = G(n), w = G(n+1)</math>, then
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| :<math>
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| \begin{matrix}
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| 23G(2n-1) &=& 4u^2 + 3v^2 + 9w^2 + 18uv - 12uw - 4vw \\
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| 23G(2n) &=& - 6u^2 + 7v^2 - 2w^2 - 4uv + 18uw + 6vw\\
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| 23G(2n+1) &=& 9u^2 + v^2 + 3w^2 + 6uv - 4uw + 14vw \\
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| 23G(3n-1)& = &\left(-4u^3 + 2v^3 -w^3 + 9(uv^2+vw^2+wu^2) + 3v^2w+6uvw\right)\\
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| 23G(3n)& = &\left(3u^3 + 2v^3 + 3w^3 - 3(uv^2 + uw^2 + vw^2 + vu^2) + 6v^2w + 18uvw\right) \\
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| 23G(3n+1)& = &\left(v^3-w^3+6uv^2+9uw^2+6vw^2+9vu^2-3wu^2+6wv^2-6uvw\right) \end{matrix}
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| </math>
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| The number 23 here arises from the discriminant of the defining polynomial of the sequence.
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| This allows you to compute the nth Perrin number using integer arithmetic in <math>O(\log n)</math> multiplies.
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| == Primes and divisibility ==
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| === Perrin pseudoprimes ===
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| It has been proven that for all primes ''p'', ''p'' divides ''P''(''p''). However, the converse is not true: for some [[composite number]]s ''n'', ''n'' may still divide ''P''(''n''). If ''n'' has this property, it is called a '''Perrin [[pseudoprime]]'''.
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| 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 = 521<sup>2</sup>; the next-smallest is 904631 = 7 x 13 x 9941. There are seventeen of them less than a billion;<ref>{{OEIS|id=A013998}}</ref> Jon Grantham has proved<ref>{{cite journal |author=[[Jon Grantham]] |title=There are infinitely many Perrin pseudoprimes |journal=[[Journal of Number Theory]] |year=2010 |volume=130 |issue=5 |pages=1117–1128 |doi=10.1016/j.jnt.2009.11.008 |url=http://www.pseudoprime.com/pseudo3.pdf}}</ref> that there are infinitely many Perrin pseudoprimes.
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| === Perrin primes ===
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| A '''Perrin prime''' is a Perrin number that is [[prime number|prime]]. The first few Perrin primes are:
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| :2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 {{OEIS|id=A074788}}
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| [[Eric W. Weisstein|E. W. Weisstein]] found a 32,147 digit probable Perrin prime ''P''(263226) in May 2006.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite journal
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| | author = Adams, William; Shanks, Daniel
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| | title = Strong primality tests that are not sufficient
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| | journal = Mathematics of Computation
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| | volume = 39
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| | year = 1982
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| | issue = 159
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| | pages = 255–300
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| | id = {{MathSciNet | id = 0658231}}
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| | doi = 10.2307/2007637
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| | jstor = 2007637
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| | publisher = American Mathematical Society}}
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| *{{cite journal
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| | author = Füredi, Z. | authorlink = Zoltán Füredi
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| | title = The number of maximal independent sets in connected graphs
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| | journal = Journal of Graph Theory
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| | volume = 11
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| | issue = 4
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| | year = 1987
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| | pages = 463–470
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| | doi = 10.1002/jgt.3190110403}}
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| * {{cite isbn | 0201038048}}
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| *{{cite journal
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| | author = Lucas, E.
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| | authorlink = Édouard Lucas
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| | title = Théorie des fonctions numériques simplement périodiques
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| | journal = American Journal of Mathematics
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| | volume = 1
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| | pages = 197–240
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| | year = 1878
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| | doi = 10.2307/2369311
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| | jstor = 2369311
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| | issue = 3
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| | publisher = The Johns Hopkins University Press}}
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| *{{cite journal
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| | author = Perrin, R.
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| | title = Query 1484
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| | journal = [[L'Intermédiaire des Mathématiciens]]
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| | volume = 6
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| | pages = 76
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| | year = 1899}}
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| ==External links==
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| *[http://www.ai.univie.ac.at/perrin.html Zentrum für Hirnforschung Institut für Medizinische Kybernetik und Artificial Intelligence]
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| *[http://www.mathpages.com/home/kmath127/kmath127.htm MathPages - Lucas Pseudoprimes]
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| *[http://www.mathpages.com/home/kmath345/kmath345.htm MathPages - Perrin's Sequence]
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| *[http://oeis.org/A225876 Perrin-like sequence]
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| {{Prime number classes}}
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| {{Classes of natural numbers}}
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| [[Category:Integer sequences]]
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| [[Category:Recurrence relations]]
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