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| {{distinguish2|the sequence of [[Lucas number]]s, which is a particular Lucas sequence}}
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| In [[mathematics]], the '''Lucas sequences''' ''U''<sub>''n''</sub>(''P'',''Q'') and ''V''<sub>''n''</sub>(''P'',''Q'') are certain [[integer sequence]]s that satisfy the [[recurrence relation]]
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| :''x''<sub>''n''</sub> = ''P x''<sub>''n''−1</sub> − ''Q x''<sub>''n''−2</sub>
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| where ''P'' and ''Q'' are fixed integers. Any other sequence satisfying this recurrence relation can be represented as a [[linear combination]] of the Lucas sequences ''U''<sub>''n''</sub>(''P'',''Q'') and ''V''<sub>''n''</sub>(''P'',''Q'').
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| More generally, Lucas sequences ''U''<sub>''n''</sub>(''P'',''Q'') and ''V''<sub>''n''</sub>(''P'',''Q'') represent sequences of polynomials in ''P'' and ''Q'' with integer coefficients.
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| Famous examples of Lucas sequences include the [[Fibonacci number]]s, [[Mersenne number]]s, [[Pell number]]s, [[Lucas number]]s, [[Jacobsthal number]]s, and a superset of [[Fermat number]]s. Lucas sequences are named after the [[France|French]] [[mathematician]] [[Édouard Lucas]].
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| == Recurrence relations ==
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| Given two integer parameters ''P'' and ''Q'', the Lucas sequences of the first kind ''U''<sub>''n''</sub>(''P'',''Q'') and of the second kind ''V''<sub>''n''</sub>(''P'',''Q'') are defined by the [[recurrence relation]]s:
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| :<math>U_0(P,Q)=0, \,</math>
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| <!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| :<math>U_1(P,Q)=1, \,</math>
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| <!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| :<math>U_n(P,Q)=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q) \mbox{ for }n>1, \,</math>
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| <!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| and
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| :<math>V_0(P,Q)=2, \,</math>
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| <!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| :<math>V_1(P,Q)=P, \,</math>
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| <!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| :<math>V_n(P,Q)=P\cdot V_{n-1}(P,Q)-Q\cdot V_{n-2}(P,Q) \mbox{ for }n>1, \,</math>
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| <!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| It is not hard to show that for <math>n>0</math>,
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| :<math>U_n(P,Q)=\frac{P\cdot U_{n-1}(P,Q) + V_{n-1}(P,Q)}{2}, \,</math>
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| :<math>V_n(P,Q)=\frac{(P^2-4Q)\cdot U_{n-1}(P,Q)+P\cdot V_{n-1}(P,Q)}{2}. \,</math>
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| == Examples ==
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| Initial terms of Lucas sequences ''U''<sub>''n''</sub>(''P'',''Q'') and ''V''<sub>''n''</sub>(''P'',''Q'') are given in the table:
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| {|class="wikitable" style="background: #fff"
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| |-
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| !<math>n\,</math>!!<math>U_n(P,Q)\, </math>!!<math>V_n(P,Q)\, </math>
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| |-
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| |<math>0\, </math>||<math>0\, </math>||<math>2\, </math>
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| |-
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| |<math>1\, </math>||<math>1\, </math>||<math>P\, </math>
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| |-
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| |<math>2\, </math>||<math>P\, </math>||<math>{P}^{2}-2Q\, </math>
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| |-
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| |<math>3\, </math>||<math>{P}^{2}-Q\, </math>||<math>{P}^{3}-3PQ\, </math>
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| |-
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| |<math>4\, </math>||<math>{P}^{3}-2PQ\, </math>||<math>{P}^{4}-4{P}^{2}Q+2{Q}^{2}\, </math>
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| |-
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| |<math>5\, </math>||<math>{P}^{4}-3{P}^{2}Q+{Q}^{2}\,</math>||<math>{P}^{5}-5{P}^{3}Q+5P{Q}^{2}\, </math>
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| |-
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| |<math>6\, </math>||<math>{P}^{5}-4{P}^{3}Q+3P{Q}^{2}\, </math>||<math>{P}^{6}-6{P}^{4}Q+9{P}^{2}{Q}^{2}-2{Q}^{3}\, </math>
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| |}
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| == Algebraic relations ==
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| The characteristic equation of the recurrence relation for Lucas sequences <math>U_n(P,Q)</math> and <math>V_n(P,Q)</math> is:
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| :<math>x^2 - Px + Q=0 \,</math>
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| <!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| It has the [[discriminant]] <math>D=P^2 - 4Q</math> and the roots:
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| :<math>a = \frac{P+\sqrt{D}}2\quad\text{and}\quad b = \frac{P-\sqrt{D}}2. \,</math>
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| Thus:
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| :<math>a + b = P\, ,</math>
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| :<math>a b = \frac{1}{4}(P^2 - D) = Q\, ,</math>
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| :<math>a - b = \sqrt{D}\, .</math>
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| Note that the sequence <math>a^n</math> and the sequence <math>b^n</math> also satisfy the recurrence relation. However these might not be integer sequences.
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| === Distinct roots ===
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| When <math>D\ne 0</math>, ''a'' and ''b'' are distinct and one quickly verifies that
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| :<math>a^n = \frac{V_n + U_n \sqrt{D}}{2}</math>
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| :<math>b^n = \frac{V_n - U_n \sqrt{D}}{2}</math>.
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| It follows that the terms of Lucas sequences can be expressed in terms of ''a'' and ''b'' as follows
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| :<math>U_n= \frac{a^n-b^n}{a-b} = \frac{a^n-b^n}{ \sqrt{D}}</math>
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| :<math>V_n=a^n+b^n \,</math>
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| <!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| ===Repeated root===
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| The case <math> D=0 </math> occurs exactly when <math> P=2S \text{ and }Q=S^2</math> for some integer ''S'' so that <math>a=b=S</math>. In this case one easily finds that
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| :<math>U_n(P,Q)=U_n(2S,S^2) = nS^{n-1}\,</math>
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| :<math>V_n(P,Q)=V_n(2S,S^2)=2S^n\,</math>.
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| ===Additional sequences having the same discriminant===
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| If the Lucas sequences <math>U_n(P, Q)</math> and <math>V_n(P, Q)</math> have | |
| discriminant <math>D = P^2 - 4Q</math>, then the sequences based on <math>P_2</math> and <math>Q_2</math> where
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| :<math> P_2 = P + 2 </math>
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| :<math> Q_2 = P + Q + 1 </math>
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| have the same discriminant: <math>P_2^2 - 4Q_2 = (P+2)^2 - 4(P + Q + 1) = P^2 - 4Q = D</math>.
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| == Other relations ==
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| The terms of Lucas sequences satisfy relations that are generalizations of those between [[Fibonacci number]]s <math>F_n=U_n(1,-1)</math> and [[Lucas number]]s <math>L_n=V_n(1,-1)</math>. For example:
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| {|class="wikitable" style="background: #fff"
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| !General!!''P'' = 1, ''Q'' = -1
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| |-
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| |<math>(P^2-4Q) U_n = {V_{n+1} - Q V_{n-1}}=2V_{n+1}-P V_n \,</math>||<math>5F_n = {L_{n+1} + L_{n-1}}=2L_{n+1} - L_{n} \,</math>
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| |-
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| |<math>V_n = U_{n+1} - Q U_{n-1}=2U_{n+1}-PU_n \,</math>||<math>L_n = F_{n+1} + F_{n-1}=2F_{n+1}-F_n \,</math>
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| |-
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| |<math>U_{2n} = U_n V_n \,</math>||<math>F_{2n} = F_n L_n \,</math>
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| |-
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| |<math>V_{2n} = V_n^2 - 2Q^n \,</math>||<math>L_{2n} = L_n^2 - 2(-1)^n \,</math>
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| |-
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| |<math>U_{n+m} = U_n U_{m+1} - Q U_m U_{n-1}=\frac{U_nV_m+U_mV_n}{2} \,</math>||<math>F_{n+m} = F_n F_{m+1} + F_m F_{n-1}=\frac{F_nL_m+F_mL_n}{2} \,</math>
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| |-
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| |<math>V_{n+m} = V_n V_m - Q^m V_{n-m} \,</math>||<math>L_{n+m} = L_n L_m - (-1)^m L_{n-m} \,</math>
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| |}
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| <!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| Among the consequences is that <math>U_{km}(P,Q)</math> is a multiple of <math>U_m(P,Q)</math>, i.e., the sequence <math>(U_m(P,Q))_{m\ge1}</math>
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| is a [[divisibility sequence]]. This implies, in particular, that <math>U_n(P,Q)</math> can be prime only when ''n'' is prime.
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| Another consequence is an analog of [[exponentiation by squaring]] that allows fast computation of <math>U_n(P,Q)</math> for large values of ''n''.
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| These facts are used in the [[Lucas–Lehmer primality test]].
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| [[Carmichael's theorem]] states that all but finitely many of the terms in a Lucas sequence have a [[prime factor]] that does not divide any earlier term in the sequence {{harv|Yubuta|2001}}.
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| == Specific names ==
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| The Lucas sequences for some values of ''P'' and ''Q'' have specific names:
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| :''U<sub>n</sub>''(1,−1) : [[Fibonacci number]]s
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| :''V<sub>n</sub>''(1,−1) : [[Lucas number]]s
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| :''U<sub>n</sub>''(2,−1) : [[Pell number]]s
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| :''V<sub>n</sub>''(2,−1) : Companion Pell numbers or Pell-Lucas numbers
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| :''U<sub>n</sub>''(1,−2) : [[Jacobsthal number]]s
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| :''V<sub>n</sub>''(1,−2) : [[Jacobsthal-Lucas numbers]] | |
| :''U<sub>n</sub>''(3, 2) : [[Mersenne number]]s 2<sup>''n''</sup> − 1
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| :''V<sub>n</sub>''(3, 2) : Numbers of the form 2<sup>''n''</sup> + 1, which include the [[Fermat number]]s {{harv|Yubuta|2001}}.
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| :''U<sub>n</sub>''(''x'',−1) : [[Fibonacci polynomials]] | |
| :''V<sub>n</sub>''(''x'',−1) : [[Lucas polynomials]].
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| Some Lucas sequences have entries in the [[On-Line Encyclopedia of Integer Sequences]]:
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| :{|class="wikitable" style="background: #fff"
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| !<math>P\,</math>!!<math>Q\, </math>!!<math>U_n(P,Q)\, </math>!! <math>V_n(P,Q)\,</math>
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| |-
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| | -1 || 3 || {{OEIS2C|A214733}}
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| | 1 || -1 || {{OEIS2C|A000045}} || {{OEIS2C|A000032}}
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| |-
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| | 1 || 1 || {{OEIS2C|A128834}} || {{OEIS2C|A087204}}
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| |-
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| | 1 || 2 || {{OEIS2C|A107920}}
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| |-
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| | 2 || -1 || {{OEIS2C|A000129}} || {{OEIS2C|A002203}}
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| |-
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| | 2 || 1 || {{OEIS2C|A001477}}
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| |-
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| | 2 || 2 || {{OEIS2C|A009545}} || {{OEIS2C|A007395}}
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| |-
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| | 2 || 3 || {{OEIS2C|A088137}}
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| |-
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| | 2 || 4 || {{OEIS2C|A088138}}
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| |-
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| | 2 || 5 || {{OEIS2C|A045873}}
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| |-
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| | 3 || -5 || {{OEIS2C|A015523}} || {{OEIS2C|A072263}}
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| |-
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| | 3 || -4 || {{OEIS2C|A015521}} || {{OEIS2C|A201455}}
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| |-
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| | 3 || -3 || {{OEIS2C|A030195}} || {{OEIS2C|A172012}}
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| |-
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| | 3 || -2 || || {{OEIS2C|A206776}}
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| |-
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| | 3 || -1 || {{OEIS2C|A006190}} || {{OEIS2C|A006497}}
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| |-
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| | 3 || 1 || {{OEIS2C|A001906}} || {{OEIS2C|A005248}}
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| |-
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| | 3 || 2 || {{OEIS2C|A000225}} || {{OEIS2C|A000051}}
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| |-
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| | 3 || 5 || {{OEIS2C|A190959}}
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| |-
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| | 4 || -3 || {{OEIS2C|A015530}} || {{OEIS2C|A080042}}
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| |-
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| | 4 || -2 || {{OEIS2C|A090017}}
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| |-
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| | 4 || -1 || {{OEIS2C|A001076}} || {{OEIS2C|A014448}}
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| |-
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| | 4 || 1 || {{OEIS2C|A001353}} || {{OEIS2C|A003500}}
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| |-
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| | 4 || 2 || || {{OEIS2C|A056236}}
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| |-
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| | 4 || 3 || {{OEIS2C|A003462}} || {{OEIS2C|A034472}}
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| |-
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| | 4 || 4 || {{OEIS2C|A001787}}
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| |-
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| | 5 || -3 ||{{OEIS2C|A015536}}
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| |-
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| | 5 || -2 ||{{OEIS2C|A015535}}
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| |-
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| | 5 || -1 || || {{OEIS2C|A087130}}
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| |-
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| | 5 || 1 || || {{OEIS2C|A003501}}
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| |-
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| | 5 || 4 ||{{OEIS2C|A002450}} || {{OEIS2C|A052539}}
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| |}
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| ==Applications==
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| * LUC is a [[public-key cryptosystem]] based on Lucas sequences<ref>{{cite journal |author=P. J. Smith, M. J. J. Lennon |title=LUC: A new public key system |journal=Proceedings of the Ninth IFIP Int. Symp. on Computer Security |year=1993 |pages=103–117 |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.1835}}</ref> that implements the analogs of [[ElGamal]] (LUCELG), [[Diffie-Hellman]] (LUCDIF), and [[RSA (algorithm)|RSA]] (LUCRSA). The encryption of the message in LUC is computed as a term of certain Lucas sequence, instead of using [[modular exponentiation]] as in RSA or Diffie-Hellman. However, a paper by Bleichenbacher et al.<ref>{{cite journal |author=D. Bleichenbacher, W. Bosma, A. K. Lenstra |title=Some Remarks on Lucas-Based Cryptosystems |journal=[[Lecture Notes in Computer Science]] |volume=963 |year=1995 |pages=386–396 |doi=10.1007/3-540-44750-4_31 |url=http://www.math.ru.nl/~bosma/pubs/CRYPTO95.pdf}}</ref> shows that many of the supposed security advantages of LUC over cryptosystems based on modular exponentiation are either not present, or not as substantial as claimed.
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| * Lucas sequences are used in probabilistic [[Lucas pseudoprime]] tests.
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| ==References==
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| {{reflist}}
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| * {{cite journal| first1=D. H. | last1=Lehmer
| |
| |title=An extended theory of Lucas' functions
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| |journal=Annals of Mathematics |year=1930
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| |volume=31 | number=3
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| |jstor=1968235 |pages=419–448 |bibcode=1930AnMat..31..419L
| |
| }}
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| *{{cite journal| first1=Morgan | last1=Ward
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| |title=Prime divisors of second order recurring sequences
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| |journal = Duke Math. J. | year=1954 | volume=21 | number=4
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| |pages=607–614 | mr=0064073 |doi=10.1215/S0012-7094-54-02163-8
| |
| }}
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| *{{cite journal|first1=Lawrence | last1=Somer
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| |title=The divisibility properties of primary Lucas Recurrences with respect to primes
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| |year=1980 | journal=Fib. Quart. | pages=316 | volume=18 | url=http://www.fq.math.ca/Scanned/18-4/somer.pdf
| |
| }}
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| * {{cite journal|first1=J. C. | last1=Lagarias
| |
| |journal=Pac. J. Math | title=The set of primes dividing Lucas Numbers has density 2/3
| |
| |year=1985 | volume=118 | number=2 | pages=449–461 | mr=789184
| |
| }}
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| * {{cite book | title=Prime Numbers and Computer Methods for Factorization | edition=2nd ed | author=Hans Riesel | authorlink=Hans Riesel | series=Progress in Mathematics | volume=126 | publisher=Birkhäuser | year=1994 | isbn=0-8176-3743-5 | pages=107–121 }}
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| * {{ cite journal|first1=Paulo | last1=Ribenboim | first2=Wayne L. |last2=McDaniel
| |
| |title=The square terms in Lucas Sequences | journal=J. Numb. Theory
| |
| |year=1996 | volume=58 | number=1 | pages=104–123 | doi=10.1006/jnth.1996.0068
| |
| }}
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| * {{cite journal | first1=M. | last1=Joye | first2=J.-J. | last2=Quisquater
| |
| |title=Efficient computation of full Lucas sequences
| |
| |journal=El. Lett. | year=1996 | volume=32|number=6 | pages=537–538
| |
| ||url=http://www.joye.site88.net/papers/JQ96lucas.pdf |doi=10.1049/el:19960359
| |
| }}
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| *{{cite book | first=Paulo | last=Ribenboim | authorlink=Paulo Ribenboim | coauthors= | year=2000 | title=My Numbers, My Friends: Popular Lectures on Number Theory | edition= | publisher=[[Springer-Verlag]] | location=New York | isbn=0-387-98911-0 | pages=1–50 }}
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| *{{cite journal | first1=Florian | last1=Luca
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| |title=Perfect Fibonacci and Lucas numbers | year=2000
| |
| |journal = Rend. Circ Matem. Palermo
| |
| |doi=10.1007/BF02904236 | volume=49 | number=2 | pages=313–318
| |
| }}
| |
| *{{cite journal
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| | last = Yabuta | first = M.
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| | journal = Fibonacci Quarterly
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| | pages = 439–443
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| | title = A simple proof of Carmichael's theorem on primitive divisors
| |
| | url = http://www.fq.math.ca/Scanned/39-5/yabuta.pdf
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| | volume = 39
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| | year = 2001}}.
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| *{{cite book | title=Proofs that Really Count | author=Arthur T. Benjamin | coauthors=Jennifer J. Quinn | publisher=[[Mathematical Association of America]] | year=2003 | isbn=0-88385-333-7 | page=35 }}
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| *{{MathWorld | urlname=LucasSequence | title=Lucas Sequence}}
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| * {{cite web| first1=Wei | last1= Dai | url = http://weidai.com/lucas.html
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| |title= Lucas Sequences in Cryptography}}
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| ==See also==
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| *[[Somer–Lucas pseudoprime]]
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| [[Category:Recurrence relations]]
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| [[Category:Integer sequences]]
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