Lucas Number Calculator

L(n) = F(n−1)+F(n+1). Also F(2n) = F(n)·L(n). They satisfy the same recurrence with different initial conditions. Together they form a 'Lucas pair' that satisfies many beautiful polynomial identities.

For Mersenne numbers Mₚ = 2ᵖ−1: define s₀=4, sₙ₊₁=sₙ²−2. Then Mₚ is prime iff sₚ₋₂ ≡ 0 (mod Mₚ). This Lucas-Lehmer test is the fastest known primality test for Mersenne numbers.

While F(n) = (φⁿ−ψⁿ)/√5 (difference), L(n) = φⁿ+ψⁿ (sum). Since |ψ|<1, ψⁿ→0, so L(n) ≈ φⁿ for large n. The initial conditions L(0)=2=φ⁰+ψ⁰ and L(1)=1=φ+ψ confirm this.