Editing Lucas pseudoprime (section)

==Fibonacci pseudoprimes==
When ''P'' = 1 and ''Q'' = &minus;1, the ''U<sub>n</sub>''(''P'',''Q'') sequence represents the Fibonacci numbers.

A '''Fibonacci pseudoprime''' is often<ref name="Bressoud"/>{{rp|264,}}<ref name="CrandallPomerance"/>{{rp|142,}}<ref name="Ribenboim"/>{{rp|127}}
defined as a composite number ''n'' not divisible by 5 for which congruence ({{EquationNote|1}}) holds with ''P'' = 1 and ''Q'' = &minus;1. By this definition, the Fibonacci pseudoprimes form a sequence:
: 323, 377, 1891, 3827, 4181, 5777, 6601, 6721, 8149, 10877, ... {{OEIS|id=A081264}}.  
The references of Anderson and Jacobsen below use this definition.

If ''n'' is congruent to 2 or 3 modulo 5, then Bressoud,<ref name="Bressoud"/>{{rp|272–273}} and Crandall and Pomerance<ref name="CrandallPomerance"/>{{rp|143,168}} point out that it is rare for a Fibonacci pseudoprime to also be a [[Fermat pseudoprime]] base 2.  However, when ''n'' is congruent to 1 or 4 modulo 5, the opposite is true, with over 12% of Fibonacci pseudoprimes under 10<sup>11</sup> also being base-2 Fermat pseudoprimes.

If ''n'' is prime and GCD(''n'', ''Q'') = 1, then we also have<ref name="lpsp"/>{{rp|1392}}
{{NumBlk|:|<math>V_n(P,Q) \equiv P \pmod{n}.</math>|{{EquationRef|5}}}}

This leads to an alternative definition of Fibonacci pseudoprime:<ref name = FibonacciPSP>{{cite journal |author1 = Adina Di Porto |author2 = Piero Filipponi |title = More on the Fibonacci Pseudoprimes |journal = Fibonacci Quarterly |date = 1989 |volume = 27 |issue = 3 |pages = 232–242 |url = https://www.fq.math.ca/Issues/27-3.pdf}}</ref><ref>{{cite journal |author=Di Porto, Adina |author2=Filipponi, Piero |author3=Montolivo, Emilio |title=On the generalized Fibonacci pseudoprimes |journal=Fibonacci Quarterly |volume=28 |pages=347–354 |date=1990 |citeseerx=10.1.1.388.4993}}</ref>
: a '''Fibonacci pseudoprime''' is a composite number ''n'' for which congruence ({{EquationNote|5}}) holds with ''P'' = 1 and ''Q'' = &minus;1.
This definition leads the Fibonacci pseudoprimes form a sequence:
: 705, 2465, 2737, 3745, 4181, 5777, 6721, 10877, 13201, 15251, ... {{OEIS|id=A005845}},
which are also referred to as ''Bruckman-Lucas'' pseudoprimes.<ref name="Ribenboim"/>{{rp|129}}
Hoggatt and Bicknell studied properties of these pseudoprimes in 1974.<ref>{{cite journal |author1 = V. E. Hoggatt, Jr. |author-link1 = Verner Emil Hoggatt Jr. |author2 = Marjorie Bicknell |title = Some Congruences of the Fibonacci Numbers Modulo a Prime p |journal = Mathematics Magazine |date = September 1974 |volume = 47 |issue = 4 |pages = 210–214 |doi = 10.2307/2689212 |jstor = 2689212}}</ref> Singmaster computed these pseudoprimes up to 100000.<ref>{{cite journal |author = David Singmaster |title=Some Lucas Pseudoprimes |journal=Abstracts Amer. Math. Soc. |date=1983 |volume=4 |issue=83T–10–146 |page=197}}</ref> Jacobsen lists all 111443 of these pseudoprimes less than 10<sup>13</sup>.<ref>{{cite web |title = Pseudoprime Statistics and Tables |url = http://ntheory.org/pseudoprimes.html |access-date = 5 May 2019}}</ref>

It has been shown that there are no even Fibonacci pseudoprimes as defined by equation (5).<ref name = Bruckman>{{cite journal |author = P. S. Bruckman |title = Lucas Pseudoprimes are odd |journal = Fibonacci Quarterly |date = 1994 |volume = 32 |pages = 155–157}}</ref><ref>{{cite journal|author=Di Porto, Adina|title=Nonexistence of Even Fibonacci Pseudoprimes of the First Kind|journal=Fibonacci Quarterly|volume=31|pages=173–177|date=1993|citeseerx=10.1.1.376.2601}}</ref> However, even Fibonacci pseudoprimes do exist {{OEIS|id=A141137}} under the first definition given by ({{EquationNote|1}}).

A '''strong Fibonacci pseudoprime''' is a composite number ''n'' for which congruence ({{EquationNote|5}}) holds for ''Q'' = &minus;1 and all ''P''.<ref name="Muller">{{cite conference|author=Müller, Winfried B.|author2=Oswald, Alan|title=Generalized Fibonacci Pseudoprimes and Probable Primes|editor=G.E. Bergum|book-title=Applications of Fibonacci Numbers|volume=5|pages=459–464|publisher=Kluwer|date=1993|doi=10.1007/978-94-011-2058-6_45|display-editors=etal}}</ref>  It follows<ref name="Muller"/>{{rp|460}} that an odd composite integer ''n'' is a strong Fibonacci pseudoprime if and only if:
#''n'' is a [[Carmichael number]]
#2(''p'' + 1) | (''n'' &minus; 1) or 2(''p'' + 1) | (''n'' &minus; ''p'') for every prime ''p'' dividing ''n''.

The smallest example of a strong Fibonacci pseudoprime is 443372888629441 = 17·31·41·43·89·97·167·331.