Lucas number (another Prime Pages' Glossary entries)
 Glossary: Prime Pages: Top 5000: 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, vn+1 = vn + vn-1 which is very similar to the recurrence relation for the Fibonacci numbers: u1 = 1, u2 = 1, un+1 = un + un-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: vn divides vmn if and only if m is an odd number vn2 - vn+1 vn-1 = 5 (-1)n v2n = vn2 - 2 (-1)n vn = u2n / un vn = un-1 + un+1 This page contributed by T. D. Noe See Also: FibonacciNumber, FibonacciPrime, LucasPrimeRelated pages (outside of this work)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.] Chris K. Caldwell © 1999-2018 (all rights reserved)