Lucas number
(another Prime Pages' Glossary entries)
The Prime Glossary
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, LucasPrime

Related 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.]



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