# Lucas number

Lucas numbers, named after French mathematician Édouard Lucas (1842-1891), are numbers in the sequence 1,3,4,7,11,18,29,... defined by the recurrence relationwhich is very similar to the recurrence relation for the Fibonacci numbers:v_{1}= 1,v_{2}= 3,v_{n+1}=v+_{n}v_{n-1}

In fact, it was Edouard Lucas who gave the Fibonacci sequence its name. Lucas and Fibonacci numbers satisfy many interesting identities, a few of which are given here:u_{1}= 1,u_{2}= 1,u_{n+1}=u+_{n}u_{n-1}

*v*divides_{n}*v*if and only if_{mn}*m*is an odd number*v*_{n}^{2}-*v*_{n+1}*v*_{n-1}= 5 (-1)^{n}*v*_{2n}=*v*_{n}^{2}- 2 (-1)^{n}*v*=_{n}*u*_{2n}/*u*_{n}*v*=_{n}*u*_{n-1}+*u*_{n+1}

This page contributed by T. D. Noe

**See Also:** FibonacciNumber, FibonacciPrime, LucasPrime

**Related pages** (outside of this work)

**References:**

- BMS88
J. Brillhart,P. L. MontgomeryandR. D. Silverman, "Tables of Fibonacci and Lucas factorizations,"Math. Comp.,50(1988) 251--260, S1--S15.MR 89h:11002[See also [DK99].]- DK99
H. DubnerandW. Keller, "New Fibonacci and Lucas primes,"Math. Comp.,68:225 (1999) 417--427, S1--S12.MR 99c:11008[Probable primality ofF,L,F*andL*tested fornup to 50000, 50000, 20000, and 15000, respectively. Many new primes and algebraic factorizations found.]

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