Lucas sequence

 ( Recurrence Relations ) 

In mathematics, the Lucas sequences $U\_n(P,Q)$ and $V\_n(P,\; Q)$ are certain constant-recursive that satisfy the recurrence relation

$x\_n\; =\; P\; \backslash cdot\; x\_\{n\; -\; 1\}\; -\; Q\; \backslash cdot\; x\_\{n\; -\; 2\}$

where $P$ and $Q$ are fixed . Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences $U\_n(P,\; Q)$ and $V\_n(P,\; Q).$

More generally, Lucas sequences $U\_n(P,\; Q)$ and $V\_n(P,\; Q)$ represent sequences of in $P$ and $Q$ with integer .

Famous examples of Lucas sequences include the , , , , Jacobsthal numbers, and a superset of (see below). Lucas sequences are named after the France mathematician Édouard Lucas.

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Recurrence relations Given two integer parameters $P$ and $Q$, the Lucas sequences of the first kind $U\_n(P,Q)$ and of the second kind $V\_n(P,Q)$ are defined by the recurrence relations:

$\backslash begin\{align\}$

U_0(P,Q)&=0, \\ U_1(P,Q)&=1, \\ U_n(P,Q)&=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q) \mbox{ for }n>1, \end{align}

and

$\backslash begin\{align\}$

V_0(P,Q)&=2, \\ V_1(P,Q)&=P, \\ V_n(P,Q)&=P\cdot V_{n-1}(P,Q)-Q\cdot V_{n-2}(P,Q) \mbox{ for }n>1. \end{align}

It is not hard to show that for $n>0$,

$\backslash begin\{align\}$

U_n(P,Q)&=\frac{P\cdot U_{n-1}(P,Q) + V_{n-1}(P,Q)}{2}, \\ V_n(P,Q)&=\frac{(P^2-4Q)\cdot U_{n-1}(P,Q)+P\cdot V_{n-1}(P,Q)}{2}. \end{align}

The above relations can be stated in matrix form as follows:

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Examples Initial terms of Lucas sequences $U\_n(P,Q)$ and $V\_n(P,Q)$ are given in the table:

$$

\begin{array}{r|l|l} n & U_n(P,Q) & V_n(P,Q) \\ \hline 0 & 0 & 2 \\ 1 & 1 & P \\ 2 & P & {P}^{2}-2Q \\ 3 & {P}^{2}-Q & {P}^{3}-3PQ \\ 4 & {P}^{3}-2PQ & {P}^{4}-4{P}^{2}Q+2{Q}^{2} \\ 5 & {P}^{4}-3{P}^{2}Q+{Q}^{2} & {P}^{5}-5{P}^{3}Q+5P{Q}^{2} \\ 6 & {P}^{5}-4{P}^{3}Q+3P{Q}^{2} & {P}^{6}-6{P}^{4}Q+9{P}^{2}{Q}^{2}-2{Q}^{3} \end{array}

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Explicit expressions The characteristic equation of the recurrence relation for Lucas sequences $U\_n(P,Q)$ and $V\_n(P,Q)$ is:

$x^2\; -\; Px\; +\; Q=0\; \backslash ,$

It has the discriminant $D\; =\; P^2\; -\; 4Q$ and the roots:

$a\; =\; \backslash frac\{P+\backslash sqrt\{D\}\}2\backslash quad\backslash text\{and\}\backslash quad\; b\; =\; \backslash frac\{P-\backslash sqrt\{D\}\}2.\; \backslash ,$

Thus:

$a\; +\; b\; =\; P\backslash ,\; ,$

$a\; b\; =\; \backslash frac\{1\}\{4\}(P^2\; -\; D)\; =\; Q\backslash ,\; ,$

$a\; -\; b\; =\; \backslash sqrt\{D\}\backslash ,\; .$

Note that the sequence $a^n$ and the sequence $b^n$ also satisfy the recurrence relation. However these might not be integer sequences.

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Distinct roots When $D\backslash ne\; 0$, a and b are distinct and one quickly verifies that

$a^n\; =\; \backslash frac\{V\_n\; +\; U\_n\; \backslash sqrt\{D\}\}\{2\}$

$b^n\; =\; \backslash frac\{V\_n\; -\; U\_n\; \backslash sqrt\{D\}\}\{2\}.$

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

$U\_n\; =\; \backslash frac\{a^n-b^n\}\{a-b\}\; =\; \backslash frac\{a^n-b^n\}\{\; \backslash sqrt\{D\}\}$

$V\_n\; =\; a^n+b^n\; \backslash ,$

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

$U\_n(P,Q)=U\_n(2S,S^2)\; =\; nS^\{n-1\}\backslash ,$

$V\_n(P,Q)=V\_n(2S,S^2)=2S^n.\backslash ,$

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Properties

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Generating functions The ordinary generating functions are

$$

\sum_{n\ge 0} U_n(P,Q)z^n = \frac{z}{1-Pz+Qz^2};

$$

\sum_{n\ge 0} V_n(P,Q)z^n = \frac{2-Pz}{1-Pz+Qz^2}.

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Pell equations When $Q=\backslash pm\; 1$, the Lucas sequences $U\_n(P,\; Q)$ and $V\_n(P,\; Q)$ satisfy certain :

$V\_n(P,1)^2\; -\; D\backslash cdot\; U\_n(P,1)^2\; =\; 4,$

$V\_n(P,-1)^2\; -\; D\backslash cdot\; U\_n(P,-1)^2\; =\; 4(-1)^n.$

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Relations between sequences with different parameters

• For any number c, the sequences $U\_n(P\text{'},\; Q\text{'})$ and $V\_n(P\text{'},\; Q\text{'})$ with

:$P\text{'}\; =\; P\; +\; 2c$

:$Q\text{'}\; =\; cP\; +\; Q\; +\; c^2$

have the same discriminant as $U\_n(P,\; Q)$ and $V\_n(P,\; Q)$:

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

• For any number c, we also have

: $U\_n(cP,c^2Q)\; =\; c^\{n-1\}\backslash cdot\; U\_n(P,Q),$

: $V\_n(cP,c^2Q)\; =\; c^n\backslash cdot\; V\_n(P,Q).$

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Other relations The terms of Lucas sequences satisfy relations that are generalizations of those between $F\_n=U\_n(1,-1)$ and $L\_n=V\_n(1,-1)$. For example:

$$

\begin{array}{l|l|r} \text{General case} & (P,Q) = (1,-1), D = P^2 - 4Q = 5 \\ \hline D U_n = {V_{n+1} - Q V_{n-1}}=2V_{n+1}-P V_n & 5F_n = {L_{n+1} + L_{n-1}}=2L_{n+1} - L_{n} & (1) \\ V_n = U_{n+1} - Q U_{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} - Q U_m U_{n-1} = U_mV_n-Q^nU_{m-n} & F_{m+n} = F_n F_{m+1} + F_m F_{n-1} =F_mL_n-(-1)^nF_{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 - Q U_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} = D U_m U_n + Q^n V_{m-n} & L_{m+n} = L_m L_n - (-1)^n L_{m-n} = 5 F_m F_n + (-1)^n L_{m-n} & (6) \\ V_{2n} = V_n^2 - 2Q^n = D U_n^2 + 2Q^n & L_{2n} = L_n^2 - 2(-1)^n = 5 F_n^2 + 2(-1)^n & (7) \\ U_{m+n} = \frac{U_mV_n+U_nV_m}{2} & F_{m+n} = \frac{F_mL_n+F_nL_m}{2} & (8) \\ V_{m+n}=\frac{V_mV_n+DU_mU_n}{2} & L_{m+n}=\frac{L_mL_n+5F_mF_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}

Of these, (6) and (7) allow fast calculation of V independent of U in a way analogous to exponentiation by squaring. The relation $V\_\{mn\}\; =\; V\_\{m\}(P\; =\; V\_n,\; Q\; =\; Q\_n)$ (which belongs to the section above, "relations between sequences with different parameters") is also useful for this purpose.

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Divisibility properties Among the consequences is that $U\_\{km\}(P,Q)$ is a multiple of $U\_m(P,Q)$, i.e., the sequence $(U\_m(P,Q))\_\{m\backslash ge1\}$ is a divisibility sequence. This implies, in particular, that $U\_n(P,Q)$ can be prime number only when n is prime. Another consequence is an analog of exponentiation by squaring that allows fast computation of $U\_n(P,Q)$ for large values of n. Moreover, if $\backslash gcd(P,Q)=1$, then $(U\_m(P,Q))\_\{m\backslash ge1\}$ is a strong divisibility sequence.

Other divisibility properties are as follows:For such relations and divisibility properties, see , or .

• If n is an odd multiple of m, then $V\_m$ divides $V\_n$.
• Let N be an integer relatively prime to 2 Q. If the smallest positive integer r for which N divides $U\_r$ exists, then the set of n for which N divides $U\_n$ is exactly the set of multiples of r.
• If P and Q are even, then $U\_n,\; V\_n$ are always even except $U\_1$.
• If P is odd and Q is even, then $U\_n,\; V\_n$ are always odd for every $n\; >\; 0$.
• If P is even and Q is odd, then the parity of $U\_n$ is the same as n and $V\_n$ is always even.
• If P and Q are odd, then $U\_n,\; V\_n$ are even if and only if n is a multiple of 3.
• If p is an odd prime, then $U\_p\backslash equiv\backslash left(\backslash tfrac\{D\}\{p\}\backslash right),\; V\_p\backslash equiv\; P\backslash pmod\{p\}$ (see Legendre symbol).
• If p is an odd prime which divides P and Q, then p divides $U\_n$ for every $n>1$.
• If p is an odd prime which divides P but not Q, then p divides $U\_n$ if and only if n is even.
• If p is an odd prime which divides Q but not P, then p never divides $U\_n$ for any $n\; >\; 0$.
• If p is an odd prime which divides D but not PQ, then p divides $U\_n$ if and only if p divides n.
• If p is an odd prime which does not divide PQD, then p divides $U\_l$, where $l=p-\backslash left(\backslash tfrac\{D\}\{p\}\backslash right)$.

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 n relatively prime to D and dividing $U\_l$, where $l=n-\backslash left(\backslash tfrac\{D\}\{n\}\backslash right)$. 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 $U\_n$ 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 $U\_n$ has a primitive prime factor and determines all cases $U\_n$ has no primitive prime factor.

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Specific names The Lucas sequences for some values of P and Q have specific names:

:

:

:

: Pell–Lucas numbers (companion Pell numbers)

: Jacobsthal numbers

: Jacobsthal–Lucas numbers

: 2 ⁿ − 1

: Numbers of the form 2 ⁿ + 1, which include the

: The square roots of the square triangular numbers.

: Fibonacci polynomials

: Lucas polynomials

: Chebyshev polynomials of second kind

: Chebyshev polynomials of first kind multiplied by 2

: in base x

: xⁿ + 1

Some Lucas sequences have entries in the On-Line Encyclopedia of Integer Sequences:

>

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Applications

• Lucas sequences are used in probabilistic Lucas pseudoprime tests, which are part of the commonly used Baillie–PSW primality test.
• Lucas sequences are used in some primality proof methods, including the Lucas–Lehmer and Lucas–Lehmer–Riesel tests and the hybrid N−1/N+1 methods such as those in Brillhart-Lehmer-Selfridge 1975.
• LUC is a public-key cryptosystem based on Lucas sequences that implements the analogs of ElGamal (LUCELG), Diffie–Hellman (LUCDIF), and 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.
  (1995). 9783540602217 ISBN 9783540602217
  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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Software

• SageMath implements $U\_n$ and $V\_n$ as functions lucas_number1() and lucas_number2(), respectively.

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See also

• Lucas pseudoprime
• Frobenius pseudoprime
• Somer–Lucas pseudoprime

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Notes

• Hans Riesel (1994). 9780817637439, Birkhäuser. ISBN 9780817637439
• Paulo Ribenboim (1996). 9781461207597, Springer-Verlag, New York. ISBN 9781461207597
• Paulo Ribenboim (2025). 9780387989112, Springer-Verlag. ISBN 9780387989112
• (2025). 9780883853337, Mathematical Association of America. ISBN 9780883853337
• Lucas sequence at Encyclopedia of Mathematics.

Categories: Recurrence Relations, Integer Sequences

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