Editing Lucas pseudoprime (section)

==Strong Lucas pseudoprimes==
Now, factor <math>\delta(n)=n-\left(\tfrac{D}{n}\right)</math> into the form <math>d\cdot2^s</math> where <math>d</math> is odd.

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

:<math>  U_d \equiv 0 \pmod n  </math>
or
:<math>  V_{d \cdot 2^r} \equiv 0 \pmod n  </math>

for some 0 &le; ''r'' &lt; ''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.
Therefore, the '''strong''' test is a more stringent primality test than equation ({{EquationNote|1}}).

There are infinitely many strong Lucas pseudoprimes, and therefore, infinitely many Lucas pseudoprimes.
Theorem 7 in <ref name="lpsp"/> states: Let <math>P</math> and <math>Q</math> be relatively prime positive integers for which <math>P^2 - 4Q</math> is positive but not a square. Then there is a positive constant <math>c</math> (depending on <math>P</math> and <math>Q</math>) such that the number of strong Lucas pseudoprimes not exceeding <math>x</math> is greater than <math>c \cdot \log x</math>, for <math>x</math> sufficiently large.

We can set ''Q'' = −1, then <math>U_n</math> and <math>V_n</math> are ''P''-Fibonacci sequence and ''P''-Lucas sequence, the pseudoprimes can be called '''strong Lucas pseudoprime in base ''P''''', for example, the least strong Lucas pseudoprime with ''P'' = 1, 2, 3, ... are 4181, 169, 119, ...

An '''extra strong Lucas pseudoprime'''<ref name=ExtraStrong>{{cite tech report
 |first1=Zheiyu |last1=Mo  |first2=James P. |last2=Jones
 |title=A new primality test using Lucas sequences
 |type=preprint
}} cited in {{cite journal
 |title=A Probable Prime Test With High Confidence
 |first=Jon |last=Grantham
 |journal=Journal of Number Theory
 |year=1998 |volume=72 |issue=1 |pages=32–47 |article-number=NT982247
 |doi=10.1006/jnth.1998.2247
 |arxiv=1903.06823 |citeseerx=10.1.1.56.8827 |s2cid=119640473 
 |url=https://les-mathematiques.net/vanilla/uploads/editor/ix/sp1k717k3k9a.pdf <!-- |url=http://www.pseudoprime.com/pseudo2.ps-->
}}</ref><ref name="FrobeniusPSP">{{cite journal
 |title=Frobenius Pseudoprimes
 |first=Jon |last=Grantham
 |journal=Mathematics of Computation |year=2001 |volume=70 |issue=234 |pages=873–891
 |doi=10.1090/S0025-5718-00-01197-2 |doi-access=free
 |arxiv=1903.06820 |mr=1680879 |bibcode=2001MaCom..70..873G
 |url=http://www.pseudoprime.com/pseudo1.ps
}}</ref>
is a strong Lucas pseudoprime for a set of parameters (''P'', ''Q'') where ''Q'' = 1, satisfying one of the conditions

:<math>  U_d \equiv 0  \text{  and  } V_d \equiv \pm 2 \pmod {n}  </math>
or
:<math>  V_{d \cdot 2^r} \equiv 0 \pmod {n}  </math>

for some <math> 0 \le r < s-1</math>. An extra strong Lucas pseudoprime is also a strong Lucas pseudoprime for the same <math>(P,Q)</math> pair.  No number can be a strong Lucas pseudoprime to more than {{frac|4}} of all bases, or an extra strong Lucas pseudoprime to more than {{frac|8}} of all bases.