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 relation

v₁ = 1, v₂ = 3, v_n+1 = v_n + v_n-1

which is very similar to the recurrence relation for the Fibonacci numbers:

u₁ = 1, u₂ = 1, u_n+1 = u_n + u_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:

• v_n divides v_mn if and only if m is an odd number
• v_n² - v_n+1 v_n-1 = 5 (-1)ⁿ
• v_2n = v_n² - 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)

• OEIS sequence A000032

References:

BMS88

J. Brillhart, P. L. Montgomery and R. D. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251--260, S1--S15.  MR 89h:11002 [See also [DK99].]

DK99

H. Dubner and W. Keller, "New Fibonacci and Lucas primes," Math. Comp., 68:225 (1999) 417--427, S1--S12.  MR 99c:11008 [Probable primality of F, L, F* and L* tested for n up to 50000, 50000, 20000, and 15000, respectively. Many new primes and algebraic factorizations found.]