Lucas sequence

Distinct roots

When ${\displaystyle D\neq 0}$, a and b are distinct and one quickly verifies that

${\displaystyle a^{n}={\frac {V_{n}+U_{n}{\sqrt {D}}}{2}}}$

${\displaystyle b^{n}={\frac {V_{n}-U_{n}{\sqrt {D}}}{2}}.}$

It follows that the terms of Lucas sequences can be expressed in terms of a and b as follows

${\displaystyle U_{n}={\frac {a^{n}-b^{n}}{a-b}}={\frac {a^{n}-b^{n}}{\sqrt {D}}}}$

${\displaystyle V_{n}=a^{n}+b^{n}\,}$

Repeated root

The case ${\displaystyle D=0}$ occurs exactly when ${\displaystyle P=2S{\text{ and }}Q=S^{2}}$ for some integer S so that ${\displaystyle a=b=S}$. In this case one easily finds that

${\displaystyle U_{n}(P,Q)=U_{n}(2S,S^{2})=nS^{n-1}\,}$

${\displaystyle V_{n}(P,Q)=V_{n}(2S,S^{2})=2S^{n}.\,}$

Generating functions

The ordinary generating functions are

${\displaystyle \sum _{n\geq 0}U_{n}(P,Q)z^{n}={\frac {z}{1-Pz+Qz^{2}}};}$

${\displaystyle \sum _{n\geq 0}V_{n}(P,Q)z^{n}={\frac {2-Pz}{1-Pz+Qz^{2}}}.}$

Pell equations

When ${\displaystyle Q=\pm 1}$, the Lucas sequences ${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q)}$ satisfy certain Pell equations:

${\displaystyle V_{n}(P,1)^{2}-D\cdot U_{n}(P,1)^{2}=4,}$

${\displaystyle V_{n}(P,-1)^{2}-D\cdot U_{n}(P,-1)^{2}=4(-1)^{n}.}$

Relations between sequences with different parameters

• For any number c, the sequences ${\displaystyle U_{n}(P',Q')}$ and ${\displaystyle V_{n}(P',Q')}$ with

${\displaystyle P'=P+2c}$

${\displaystyle Q'=cP+Q+c^{2}}$

have the same discriminant as ${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q)}$:

${\displaystyle P'^{2}-4Q'=(P+2c)^{2}-4(cP+Q+c^{2})=P^{2}-4Q=D.}$

• For any number c, we also have

${\displaystyle U_{n}(cP,c^{2}Q)=c^{n-1}\cdot U_{n}(P,Q),}$

${\displaystyle V_{n}(cP,c^{2}Q)=c^{n}\cdot V_{n}(P,Q).}$

Other relations

The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci numbers ${\displaystyle F_{n}=U_{n}(1,-1)}$ and Lucas numbers ${\displaystyle L_{n}=V_{n}(1,-1)}$. For example:

${\displaystyle {\begin{array}{l|l|r}{\text{General case}}&(P,Q)=(1,-1),D=P^{2}-4Q=5\\\hline DU_{n}={V_{n+1}-QV_{n-1}}=2V_{n+1}-PV_{n}&5F_{n}={L_{n+1}+L_{n-1}}=2L_{n+1}-L_{n}&(1)\\V_{n}=U_{n+1}-QU_{n-1}=2U_{n+1}-PU_{n}&L_{n}=F_{n+1}+F_{n-1}=2F_{n+1}-F_{n}&(2)\\U_{m+n}=U_{n}U_{m+1}-QU_{m}U_{n-1}=U_{m}V_{n}-Q^{n}U_{m-n}&F_{m+n}=F_{n}F_{m+1}+F_{m}F_{n-1}=F_{m}L_{n}-(-1)^{n}F_{m-n}&(3)\\U_{2n}=U_{n}(U_{n+1}-QU_{n-1})=U_{n}V_{n}&F_{2n}=F_{n}(F_{n+1}+F_{n-1})=F_{n}L_{n}&(4)\\U_{2n+1}=U_{n+1}^{2}-QU_{n}^{2}&F_{2n+1}=F_{n+1}^{2}+F_{n}^{2}&(5)\\V_{m+n}=V_{m}V_{n}-Q^{n}V_{m-n}=DU_{m}U_{n}+Q^{n}V_{m-n}&L_{m+n}=L_{m}L_{n}-(-1)^{n}L_{m-n}=5F_{m}F_{n}+(-1)^{n}L_{m-n}&(6)\\V_{2n}=V_{n}^{2}-2Q^{n}=DU_{n}^{2}+2Q^{n}&L_{2n}=L_{n}^{2}-2(-1)^{n}=5F_{n}^{2}+2(-1)^{n}&(7)\\U_{m+n}={\frac {U_{m}V_{n}+U_{n}V_{m}}{2}}&F_{m+n}={\frac {F_{m}L_{n}+F_{n}L_{m}}{2}}&(8)\\V_{m+n}={\frac {V_{m}V_{n}+DU_{m}U_{n}}{2}}&L_{m+n}={\frac {L_{m}L_{n}+5F_{m}F_{n}}{2}}&(9)\\V_{n}^{2}-DU_{n}^{2}=4Q^{n}&L_{n}^{2}-5F_{n}^{2}=4(-1)^{n}&(10)\\U_{n}^{2}-U_{n-1}U_{n+1}=Q^{n-1}&F_{n}^{2}-F_{n-1}F_{n+1}=(-1)^{n-1}&(11)\\V_{n}^{2}-V_{n-1}V_{n+1}=DQ^{n-1}&L_{n}^{2}-L_{n-1}L_{n+1}=5(-1)^{n-1}&(12)\\2^{n-1}U_{n}={n \choose 1}P^{n-1}+{n \choose 3}P^{n-3}D+\cdots &2^{n-1}F_{n}={n \choose 1}+5{n \choose 3}+\cdots &(13)\\2^{n-1}V_{n}=P^{n}+{n \choose 2}P^{n-2}D+{n \choose 4}P^{n-4}D^{2}+\cdots &2^{n-1}L_{n}=1+5{n \choose 2}+5^{2}{n \choose 4}+\cdots &(14)\end{array}}}$

Divisibility properties

Among the consequences is that ${\displaystyle U_{km}(P,Q)}$ is a multiple of ${\displaystyle U_{m}(P,Q)}$, i.e., the sequence ${\displaystyle (U_{m}(P,Q))_{m\geq 1}}$ is a divisibility sequence. This implies, in particular, that ${\displaystyle U_{n}(P,Q)}$ can be prime only when n is prime. Moreover, if ${\displaystyle \gcd(P,Q)=1}$, then ${\displaystyle (U_{m}(P,Q))_{m\geq 1}}$ is a strong divisibility sequence.

Other divisibility properties are as follows:

• If n is an odd multiple of m, then ${\displaystyle V_{m}}$ divides ${\displaystyle V_{n}}$.
• Let N be an integer relatively prime to 2Q. If the smallest positive integer r for which N divides ${\displaystyle U_{r}}$ exists, then the set of n for which N divides ${\displaystyle U_{n}}$ is exactly the set of multiples of r.
• If P and Q are even, then ${\displaystyle U_{n},V_{n}}$ are always even except ${\displaystyle U_{1}}$.
• If P is odd and Q is even, then ${\displaystyle U_{n},V_{n}}$ are always odd for every ${\displaystyle n>0}$.
• If P is even and Q is odd, then the parity of ${\displaystyle U_{n}}$ is the same as n and ${\displaystyle V_{n}}$ is always even.
• If P and Q are odd, then ${\displaystyle U_{n},V_{n}}$ are even if and only if n is a multiple of 3.
• If p is an odd prime, then ${\displaystyle U_{p}\equiv \left({\tfrac {D}{p}}\right),V_{p}\equiv P{\pmod {p}}}$ (see Legendre symbol).
• If p is an odd prime which divides P and Q, then p divides ${\displaystyle U_{n}}$ for every ${\displaystyle n>1}$.
• If p is an odd prime which divides P but not Q, then p divides ${\displaystyle U_{n}}$ if and only if n is even.
• If p is an odd prime which divides Q but not P, then p never divides ${\displaystyle U_{n}}$ for any ${\displaystyle n>0}$.
• If p is an odd prime which divides D but not PQ, then p divides ${\displaystyle U_{n}}$ if and only if p divides n.
• If p is an odd prime which does not divide PQD, then p divides ${\displaystyle U_{l}}$, where ${\displaystyle l=p-\left({\tfrac {D}{p}}\right)}$.

The last fact generalizes Fermat's little theorem. These facts are used in the Lucas–Lehmer primality test. Like Fermat's little theorem, the converse of the last fact holds often, but not always; there exist composite numbers n relatively prime to D and dividing ${\displaystyle U_{l}}$, where ${\displaystyle l=n-\left({\tfrac {D}{n}}\right)}$. Such composite numbers are called Lucas pseudoprimes.

A prime factor of a term in a Lucas sequence which does not divide any earlier term in the sequence is called primitive. Carmichael's theorem states that all but finitely many of the terms in a Lucas sequence have a primitive prime factor. Indeed, Carmichael (1913) showed that if D is positive and n is not 1, 2 or 6, then ${\displaystyle U_{n}}$ has a primitive prime factor. In the case D is negative, a deep result of Bilu, Hanrot, Voutier and Mignotte shows that if n > 30, then ${\displaystyle U_{n}}$ has a primitive prime factor and determines all cases ${\displaystyle U_{n}}$ has no primitive prime factor.