Lucas sequence

In mathematics, a Lucas sequence is a particular generalisation of sequences like the Fibonacci numbers, Lucas numbers, Pell numbers or Jacobsthal numbers. Lucas sequences have one common characteristic: they can be generated over quadratic equations of the form: $\scriptstyle x^{2}-Px+Q=0\$ with $\scriptstyle P^{2}-4Q\neq 0$.

There exist two kinds of Lucas sequences:

• Sequences $\scriptstyle U(P,Q)=(U_{n}(P,Q))_{n\geq 1}$ with $\scriptstyle U_{n}(P,Q)={\frac {a^{n}-b^{n}}{a-b}}$,
• Sequences $\scriptstyle V(P,Q)=(V_{n}(P,Q))_{n\geq 1}$ with $\scriptstyle V_{n}(P,Q)=a^{n}+b^{n}\$,

where $\scriptstyle a\$ and $b\$ are the solutions

$a={\frac {P+{\sqrt {P^{2}-4Q}}}{2}}$

and

$b={\frac {P-{\sqrt {P^{2}-4Q}}}{2}}$

of the quadratic equation $\scriptstyle x^{2}-Px+Q=0$.

Properties

• The variables $\scriptstyle a\$ and $\scriptstyle b\$, and the parameter $\scriptstyle P\$ and $\scriptstyle Q\$ are interdependent. In particular, $\scriptstyle P=a+b\$ and $\scriptstyle Q=a\cdot b.$.
• For every sequence $\scriptstyle U(P,Q)=(U_{n}(P,Q))_{n\geq 1}$ it holds that $\scriptstyle U_{0}=0\$ and $U_{1}=1$.
• For every sequence $\scriptstyle V(P,Q)=(V_{n}(P,Q))_{n\geq 1}$ is holds that $\scriptstyle V_{0}=2\$ and $V_{1}=P$.

For every Lucas sequence the following are true:

• $\scriptstyle U_{2n}=U_{n}\cdot V_{n}\$
• $\scriptstyle V_{n}=U_{n+1}-QU_{n-1}\$
• $\scriptstyle V_{2n}=V_{n}^{2}-2Q^{n}\$
• $\scriptstyle \operatorname {ggT} (U_{m},U_{n})=U_{\operatorname {ggT} (m,n)}$
• $\scriptstyle m\mid n\implies U_{m}\mid U_{n}$ for all $\scriptstyle U_{m}\neq 1$

Fibonacci numbers and Lucas numbers

The two best known Lucas sequences are the Fibonacci numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle U(1,-1)\ } and the Lucas numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle V(1,-1)\ } with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle a = \frac{1+\sqrt{5}}{2}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b = \frac{1-\sqrt{5}}{2}} .

Lucas sequences and the prime numbers

If the natural number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle p\ } is a prime number then it holds that

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle p\ } divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle U_p(P,Q)-\left(\frac Dp\right)}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle p\ } divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle V_p(P,Q)-P\ }

Fermat's Little Theorem can then be seen as a special case of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle p\ } divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (V_n(P,Q) - P)\ } because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle a^p \equiv a \pmod p} is equivalent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle V_p(a+1,a) \equiv V_1(a+1,a) \pmod p} .

The converse pair of statements that if $\scriptstyle n\$ divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle U_n(P,Q)-\left(\frac Dn\right)} then is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle n\ } a prime number and if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\ } divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle V_m(P,Q)-P\ } then is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\ } a prime number) are individually false and lead to Fibonacci pseudoprimes and Lucas pseudoprimes, respectively.

Further reading

• P. Ribenboim, The New Book of Prime Number Records (3 ed.), Springer, 1996, ISBN 0-387-94457-5.
• P. Ribenboim, My Numbers, My Friends, Springer, 2000, ISBN 0-387-98911-0.