Title: Lucas Numbers
Description: Discover the Lucas numbers, a fascinating integer sequence closely related to the Fibonacci numbers. Learn their definition, properties, golden ratio connection, and applications in number theory and cryptography.
Author: Alloya Huckfield
tags:
  - Amethyst
  - Lucas-numbers
  - Édouard-Lucas
  - Binet-formula
  - Lucas-primes
icon: LiAsterisk

lucas-numbers

Lucas Numbers

Definition
The Lucas numbers are defined by the recurrence relation:

with initial conditions:

Sequence: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, ...

Key Properties

Binet's Formula (closed-form):
Fibonacci Relation:
Golden Ratio Limit:

Applications

Primality testing (Lucas-Lehmer test)
Combinatorics (tiling problems)
Cryptography (pseudorandom generators)

sacred-geometry