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| '''Lucas pseudoprime'''s and '''Fibonacci pseudoprime'''s are [[composite number|composite]] integers that pass certain tests which all [[Prime number|prime]]s and very few composite numbers pass: in this case, criteria relative to some [[Lucas sequence]].
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| == Basic properties ==
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| Given integers ''P'' and ''Q'', where ''P'' > 0 and <math>D=P^2-4Q</math>,
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| let ''U''<sub>''k''</sub>(''P'', ''Q'') and ''V''<sub>''k''</sub>(''P'', ''Q'') be the corresponding [[Lucas sequence]]s.
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| Let ''n'' be a positive integer and let <math>\left(\tfrac{D}{n}\right)</math> be the [[Jacobi symbol]]. We define
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| : <math>\delta(n)=n-\left(\tfrac{D}{n}\right).</math>
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| If ''n'' is a [[prime number|prime]] such that the [[greatest common divisor]] of ''n'' and ''Q'' (that is, GCD(''n, Q'')) is 1, then the following congruence condition holds (see page 1391 of
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| <ref name="lpsp">{{cite journal|coauthors=[[Samuel S. Wagstaff, Jr.]]|title=Lucas Pseudoprimes|journal=Mathematics of Computation|date=October 1980|volume=35|issue=152|pages=1391–1417|url=http://mpqs.free.fr/LucasPseudoprimes.pdf|author=Robert Baillie| mr=583518| doi=10.1090/S0025-5718-1980-0583518-6 }}</ref>):
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| :<math> \text{ } (1) \text{ } U_{\delta(n)} \equiv 0 \pmod {n}. </math>
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| If this equation does ''not'' hold, then ''n'' is ''not'' prime.
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| If ''n'' is ''composite'', then this equation usually does ''not'' hold (see,<ref name="lpsp"/> page 1392). These are the key facts that make Lucas sequences useful in [[primality test]]ing.
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| Some good references are chapter 8 of the book by Bressoud and Wagon (with [[Mathematica]] code),<ref name="Bressoud">
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| {{cite book | author = [[David Bressoud]] | coauthors=[[Stan Wagon]]
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| | title = A Course in Computational Number Theory | publisher = Key College Publishing in cooperation with Springer | location = New York | year = 2000 | isbn = 978-1-930190-10-8 }}
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| </ref> pages 142-152 of the book by Crandall and Pomerance,<ref name="CrandallPomerance">
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| {{cite book | title=Prime numbers: A computational perspective | edition=2nd | author=Richard E. Crandall | authorlink=Richard Crandall | coauthors=[[Carl Pomerance]] | publisher=[[Springer-Verlag]] | year=2005 | isbn=0-387-25282-7}}
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| </ref> and pages 53–74 of the book by Ribenboim
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| .<ref name="Ribenboim"> | |
| {{cite book | title=The New Book of Prime Number Records | author=[[Paulo Ribenboim]] | publisher=[[Springer-Verlag]] | year=1996 | isbn=0-387-94457-5 }}
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| </ref>
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| ==Lucas probable primes and pseudoprimes==
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| A '''Lucas probable prime''' for a given (''P, Q'') pair is ''any'' positive integer ''n'' for which equation (1) above is true (see,<ref name="lpsp"/> page 1398).
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| A '''Lucas pseudoprime''' for a given (''P, Q'') pair is a positive ''composite'' integer ''n'' for which equation (1) is true (see,<ref name="lpsp"/> page 1391).
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| Riesel
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| (<ref name="Riesel">{{cite book | title=Prime Numbers and Computer Methods for Factorization | edition=2nd | author=[[Hans Riesel]] | series=Progress in Mathematics | volume=126 | publisher=Birkhäuser | year=1994 | isbn=0-8176-3743-5 }}.</ref>
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| page 130) adds the condition that the Jacobi symbol <math>\left(\tfrac{D}{n}\right)</math> should be −1. This not usually part of the definition, although most implementations of the Lucas primality test (such as the [[Baillie-PSW primality test]]) specifically choose ''D'' so that <math>\left(\tfrac{D}{n}\right)=-1</math>. Bressoud and Wagon (,<ref name="Bressoud"/> pages 266-269) explain why the Jacobi symbol should be −1, as well as why one gets more powerful primality tests if ''Q'' ≠ ± 1.
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| A Lucas probable prime test is most useful if we choose a value of ''D'' such that the Jacobi symbol <math>\left(\tfrac{D}{n}\right)</math> is −1
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| (see page 1401 of <ref name="lpsp"/> or page 1024 of
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| <ref name="PSW">{{cite journal|coauthors=[[John L. Selfridge]], [[Samuel S. Wagstaff, Jr.]]|title=The pseudoprimes to 25·10<sup>9</sup>|journal=Mathematics of Computation|date=July 1980|volume=35|issue=151|pages=1003–1026|url=http://www.math.dartmouth.edu/~carlp/PDF/paper25.pdf|author = [[Carl Pomerance]]| doi=10.1090/S0025-5718-1980-0572872-7 }}</ref>
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| ).
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| If <math>\left(\tfrac{D}{n}\right)=-1,</math> then equation (1) becomes
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| :<math> \text{ } (2) \text{ } U_{n+1} \equiv 0 \pmod {n}. </math>
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| If congruence (2) is false, this constitutes a proof that ''n'' is composite.
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| If congruence (2) is true, then ''n'' is a Lucas probable prime.
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| In this case, either ''n'' prime or it is a Lucas pseudoprime.
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| If congruence (2) is true, then ''n'' is ''likely'' to be prime (this justifies the term '''probable''' prime), but this does not ''prove'' that ''n'' is prime.
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| As is the case with any other probabilistic primality test, if we perform additional Lucas tests with different ''D'', ''P'' and ''Q'', then unless one of the tests proves that ''n'' is composite, we gain more confidence that ''n'' is prime.
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| Examples: If ''P'' = 1, ''Q'' = −1, and ''D'' = 5, the sequence of ''U'''s is the [[Fibonacci sequence]]: ''F<sub>0</sub> = U<sub>0</sub>'' = 0, ''F<sub>1</sub> = U<sub>1</sub>'' = 1, ''F<sub>2</sub> = U<sub>2</sub>'' = 1, ''F<sub>3</sub> = U<sub>3</sub>'' = 2, etc.
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| First, let ''n'' = 17. The Jacobi symbol <math>\left(\tfrac{5}{17}\right)</math> is −1, so δ(''n'') = 18. The 18th Fibonacci number is 2584 = 17·152 and we have
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| :<math> U_{18} = 2584 \equiv 0 \pmod {17} . </math>
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| Therefore, 17 is a Lucas probable prime for this (''P, Q'') pair. In this case 17 is prime, so it is ''not'' a Lucas pseudoprime.
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| For the next example, let ''n'' = 323. We have <math>\left(\tfrac{5}{323}\right)</math> = −1, and we can compute
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| :<math> U_{324} \equiv 0 \pmod {323}. </math>
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| However, 323 = 17·19 is not prime, so 323 is a Lucas ''pseudoprime'' for this (''P, Q'') pair.
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| In fact, 323 is the smallest pseudoprime for ''P'' = 1, ''Q'' = −1.
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| We will see below that, in order to check equation (2) for a given ''n'', we do ''not'' need to compute all of the first ''n''+1 terms in the ''U'' sequence.
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| ==Strong Lucas pseudoprimes==
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| Now, factor <math>\delta(n)</math> into the form <math>d\cdot2^s</math> where <math>d</math> is odd.
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| A '''strong Lucas pseudoprime''' for a given (''P, Q'') pair is an odd composite number ''n'' with GCD(''n, D'') = 1, satisfying one of the conditions
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| :<math> U_d \equiv 0 \pmod {n} </math>
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| or
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| :<math> V_{d \cdot 2^r} \equiv 0 \pmod {n} </math>
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| for some ''r'' < ''s''; see page 1396 of.<ref name="lpsp"/> A strong Lucas pseudoprime is also a Lucas pseudoprime (for the same (''P, Q'') pair), but the converse is not necessarily true.
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| Therefore, the '''strong''' test is a more stringent primality test than equation (1).
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| An '''extra strong Lucas pseudoprime'''
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| <ref name="FrobeniusPSP">{{cite journal|title=Frobenius Pseudoprimes
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| |journal=Mathematics of Computation|date=March 2000|volume=70|issue=234|pages=873–891
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| |author=Jon Grantham| mr=1680879| doi=10.1090/S0025-5718-00-01197-2 }}</ref>
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| is a strong Lucas pseudoprime for a set of parameters (''P'', ''Q'') where ''Q'' = 1, satisfying one the conditions
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| :<math> U_d \equiv 0 \pmod {n} \text{ and } V_d \equiv \pm 2 \pmod {n} </math>
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| or
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| :<math> V_{d \cdot 2^r} \equiv 0 \pmod {n} </math>
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| for some <math>r<s-1</math>. An extra strong Lucas pseudoprime is also a strong Lucas pseudoprime for the same <math>(P,Q)</math> pair.
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| ==Implementing a Lucas probable prime test==
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| Before embarking on a probable prime test, one usually verifies that ''n'', the number to be tested for primality, is odd, is not a perfect square, and is not divisible by any small prime less than some convenient limit.
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| In this section, we will assume <math>\left(\tfrac{D}{n}\right)=-1</math>, so that δ(''n'') = ''n'' + 1.
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| Given ''n'', one technique for choosing ''D'' is to use trial and error to find the first ''D'' in the sequence 5, −7, 9, −11, ... such that the Jacobi symbol <math>\left(\tfrac{D}{n}\right)</math> is −1.
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| (If ''D'' and ''n'' have a factor in common, then <math>\left(\tfrac{D}{n}\right)=0</math>).
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| Then set ''P'' = 1 and <math>Q=(1-D)/4</math>.
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| Once we have ''P'' and ''Q'', it is a good idea to check that ''n'' has no factors in common with ''P'' or ''Q''.
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| Given ''D'', ''P'', and ''Q'', there are recurrence relations that enable us to quickly compute <math>U_{n+1}</math> and <math>V_{n+1}</math> without having to compute all the intermediate terms;
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| see [[Lucas sequence#Other relations|Lucas sequence-Other relations]]. First, we can double the subscript from <math>k</math> to <math>2k</math> in one step using the recurrence relations
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| :<math>U_{2k}=U_k\cdot V_k</math>
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| :<math>V_{2k}=V_k^2-2Q^k</math>.
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| Next, we can increase the subscript by 1 using the recurrences
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| :<math>U_{2k+1}=(P\cdot U_{2k}+V_{2k})/2</math>
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| :<math>V_{2k+1}=(D\cdot U_{2k}+P\cdot V_{2k})/2</math>.
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| (If either of these numerators is odd, we can make it be even by increasing it by ''n'', because all of these calculations are carried out [[modular arithmetic|modulo]] ''n''.)
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| Observe that, for each term that we compute in the ''U'' sequence, we compute the corresponding term in the ''V'' sequence. As we proceed, we also compute powers of ''Q''.
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| We use the bits of the binary expansion of ''n'' + 1, starting at the leftmost bit, to determine ''which'' terms in the ''U'' sequence need to be computed.
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| For example, if ''n'' + 1 = 44 (= 101100 in binary), we compute ''U''<sub>1</sub>, ''U''<sub>2</sub>, ''U''<sub>4</sub>, ''U''<sub>5</sub>, ''U''<sub>10</sub>, ''U''<sub>11</sub>, ''U''<sub>22</sub>, and ''U''<sub>44</sub>. We also compute the same-numbered terms in the ''V'' sequence and those powers of ''Q''.
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| By the end of the calculation, we will have computed ''U<sub>n+1</sub>'', ''V<sub>n+1</sub>'', and ''Q<sup>n+1</sup>''.
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| We then check equation (2) using our known value of ''U<sub>n+1</sub>''.
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| When ''D'', ''P'', and ''Q'' are chosen as described above, the first 10 Lucas pseudoprimes are (see page 1401 of <ref name="PSW"/>):
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| 323, 377, 1159, 1829, 3827, 5459, 5777, 9071, 9179, and 10877.
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| The '''strong''' versions of the Lucas test can be implemented in a similar way.
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| When ''D'', ''P'', and ''Q'' are chosen as described above, the first 10 ''strong'' Lucas pseudoprimes are: 5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, and 58519
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| {{OEIS|id=A217255}}
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| To calculate a list of ''extra strong'' Lucas pseudoprimes, set ''Q'' = 1.
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| Then try ''P'' = 3, 4, 5, 6, ..., until a value of <math>D=P^2-4Q</math> is found so that the Jacobi symbol <math>\left(\tfrac{D}{n}\right)=-1</math>.
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| With this method for selecting ''D'', ''P'', and ''Q'', the first 10 ''extra strong'' Lucas pseudoprimes are
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| 989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059, and 72389
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| {{OEIS|id=A217719}}
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| ===Checking additional congruence conditions===
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| If we have checked that equation (2) is true, there are additional congruence conditions we can check that have almost no additional computational cost. | |
| If ''n'' happens not to be prime, these additional conditions may help discover that fact.
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| If ''n'' is an odd prime and <math>\left(\tfrac{D}{n}\right)=-1</math>, then we have the following (see equation 2 on page 1392 of <ref name="lpsp"/>):
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| :<math> \text{ } (3) \text{ } V_{n+1} \equiv 2 Q \pmod {n} . </math>
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| Although this congruence condition is not, by definition, part of the Lucas probable prime test, it is almost free to check this condition because, as noted above, the value of ''V<sub>n+1</sub>'' was computed in the process of computing ''U<sub>n+1</sub>''.
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| If either equation (2) or (3) is false, this constitutes a proof that ''n'' is not prime.
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| If ''both'' of these conditions are true, then it is even more likely that ''n'' is prime than if we had checked only equation (2).
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| If the above method for choosing ''D'' happened to set ''Q'' = −1, then we can adjust ''P'' and ''Q'' so that ''D'' and <math>\left(\tfrac{D}{n}\right)</math> remain unchanged and ''P'' = ''Q'' = 5 (see [[Lucas sequence#Algebraic relations|Lucas sequence-Algebraic relations]]).
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| If we make this adjustment, there is only ''one'' composite ''n'' < 10<sup>8</sup> for which equation (3) is true (see page 1409 and Table 6 of;<ref name="lpsp"/> this ''n'' is 913 = 11·83).
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| Here is yet another congruence condition that is true for primes and that is trivial to check.
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| First, recall that <math>Q^{n+1}</math> is computed during the calculation of <math>U_{n+1}</math>.
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| It would be easy to save the previously-computed power of <math>Q</math>, namely, <math>Q^{(n+1)/2}</math>.
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| Next, if ''n'' is prime, then, by [[Euler's criterion]],
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| :<math> Q^{(n-1)/2} \equiv \left(\tfrac{Q}{n}\right) \pmod {n} </math> .
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| (Here, <math>\left(\tfrac{Q}{n}\right)</math> is the [[Legendre symbol]]; if ''n'' is prime, this is the same as the Jacobi symbol).
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| Therefore, if ''n'' is prime, we must have
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| :<math> \text{ } (4) \text{ } Q^{(n+1)/2} \equiv Q \cdot Q^{(n-1)/2} \equiv Q \cdot \left(\tfrac{Q}{n}\right) \pmod {n} </math> .
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| The Jacobi symbol on the right side is easy to compute, so this congruence is easy to check. | |
| If this congruence does not hold, then ''n'' cannot be prime.
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| Additional congruence conditions that must be satisfied if ''n'' is prime are described in Section 6 of.<ref name="lpsp"/> If ''any'' of these conditions fails to hold, then we have proved that ''n'' is not prime.
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| == Comparison with the Miller-Rabin primality test ==
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| ''k'' applications of the [[Miller-Rabin primality test]] declare a composite ''n'' to be probably prime with a probability at most (1/4)<sup>''k''</sup>.
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| There is a similar probability estimate for the strong Lucas probable prime test.<ref>{{cite journal|title=The Rabin-Monier Theorem for Lucas Pseudoprimes|journal=Mathematics of Computation|date=April 1997|volume=66|issue=218|pages=869–881|url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.192.4789|author=F. Arnault }}</ref>
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| Aside from two trivial exceptions (see below), the fraction of (''P'',''Q'') pairs (modulo ''n'') that declare a composite ''n'' to be probably prime is at most (4/15).
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| Therefore, ''k'' applications of the strong Lucas test would declare a composite ''n'' to be probably prime with a probability at most (4/15)<sup>k</sup>.
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| There are two trivial exceptions. One is ''n'' = 9. The other is when ''n'' = ''p''(''p''+2) is the product of two [[twin prime]]s. Such an ''n'' is easy to factor, because in this case, ''n''+1 = (''p''+1)<sup>2</sup> is a perfect square. One can quickly detect perfect squares using [[Newton's method]] for square roots.
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| By combining a Lucas pseudoprime test with a [[Fermat primality test]], say, to base 2, one can obtain very powerful probabilistic tests for primality, such as the [[Baillie-PSW primality test]].
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| ==Fibonacci pseudoprimes==
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| As noted above, when ''P'' = 1 and ''Q'' = −1, the numbers in the ''U'' sequence are the Fibonacci numbers.
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| A '''Fibonacci pseudoprime''' is often
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| (page 264 of,<ref name="Bressoud"/> page 142 of,<ref name="CrandallPomerance"/> or
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| page 127 of <ref name="Ribenboim"/>)
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| defined as a composite number ''n'' for which equation (1) above is true with ''P'' = 1 and ''Q'' = −1. By this definition, the first ten Fibonacci pseudoprimes are 323, 377, 1891, 3827, 4181, 5777, 6601, 6721, 8149, and 10877 {{OEIS|id=A081264}}. More of these values, along with their factorizations, are given in the references of Anderson listed below.
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| If n is congruent to 2 or 3 (mod 5), then Bressoud (,<ref name="Bressoud"/> pages 272-273) and Crandall and Pomerance (,<ref name="CrandallPomerance"/> page 143 and exercise 3.41 on page 168) point out that it is rare for a Fibonacci pseudoprime to also be a [[Fermat pseudoprime]] base 2.
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| If ''n'' is prime and GCD(''n'', ''Q'') = 1, then (see equation 4 on page 1392 of <ref name="lpsp"/>) we also have
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| :<math> \text{ } (5) \text{ } V_n \equiv P \pmod {n} . </math>
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| This leads to an alternate definition of Fibonacci pseudoprime that is sometimes used (see
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| ,<ref name="Muller">Müller, Winfried B. and Alan Oswald. "Generalized Fibonacci Pseudoprimes and Probable Primes." In G.E. Bergum et al., eds. ''Applications of Fibonacci Numbers. Volume 5.'' Dordrecht: Kluwer, 1993.</ref> pages 459-464).
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| By this definition, a '''Fibonacci pseudoprime''' is a composite number ''n'' for which equation (5) is true with ''P'' = 1 and ''Q'' = −1.
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| Using this definition, the first ten Fibonacci pseudoprimes are 2737, 4181, 5777, 6721, 10877, 13201, 15251, 29281, 34561, and 51841.
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| A '''strong Fibonacci pseudoprime''' may be defined as a composite number for which equation (5) holds ''for all'' ''P''. It follows (see,<ref name="Muller"/> page 360) that in this case:
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| #An odd composite integer ''n'' is also a [[Carmichael number]]
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| #2(''p''<sub>''i''</sub> + 1) | (''n'' − 1) or 2(''p''<sub>''i''</sub> + 1) | (''n'' − ''p''<sub>''i''</sub>) for every prime ''p''<sub>''i''</sub> dividing ''n''.
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| The smallest example of a strong Fibonacci pseudoprime is 443372888629441 = 17·31·41·43·89·97·167·331.
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| It is conjectured that there are no even Fibonacci pseudoprimes.<ref>{{cite book |author=Somer, Lawrence |chapter=On Even Fibonacci Pseudoprimes |editor=G. E. Bergum et al. |title=Applications of Fibonacci Numbers |volume=4 |place=Dordrecht |publisher=Kluwer |year=1991 |pages=277–288}}</ref>
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| ==References==
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| {{reflist}}
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| ==External links==
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| * Anderson, Peter G. [http://www.cs.rit.edu/usr/local/pub/pga/fpp_and_entry_pts Fibonacci Pseudoprimes, their factors, and their entry points.]
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| * Anderson, Peter G. [http://www.cs.rit.edu/usr/local/pub/pga/fibonacci_pp Fibonacci Pseudoprimes under 2,217,967,487 and their factors.]
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| * {{MathWorld|urlname=FibonacciPseudoprime|title=Fibonacci Pseudoprime}}
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| * {{MathWorld|urlname=LucasPseudoprime|title=Lucas Pseudoprime}}
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| * {{MathWorld|urlname=StrongLucasPseudoprime|title=Strong Lucas Pseudoprime}}
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| * {{MathWorld|urlname=ExtraStrongLucasPseudoprime|title=Extra Strong Lucas Pseudoprime}}
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| {{Classes of natural numbers}}
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| [[Category:Fibonacci numbers]]
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| [[Category:Pseudoprimes]]
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