ϕ=1+52\phi=\cfrac{1+\sqrt5}{2}ϕ=21+5, ψ=1+52\psi=\cfrac{1+\sqrt5}{2}ψ=21+5
Fn=ϕn−ψnϕ−ψ=15(ϕn−ψn)F_n=\cfrac{\phi^n-\psi^n}{\phi-\psi}=\cfrac{1}{\sqrt5}(\phi^n-\psi^n)Fn=ϕ−ψϕn−ψn=51(ϕn−ψn)
an=15((1+52)n−(1−52)n)a_n=\cfrac{1}{\sqrt5}( \left(\cfrac{1+\sqrt5}{2}\right)^n - \left(\cfrac{1-\sqrt5}{2}\right)^n)an=51((21+5)n−(21−5)n)
y2−xy−x2=±1y^2-xy-x^2=\pm1y2−xy−x2=±1
5x2+45x^2+45x2+4 또는 5x2−45x^2-45x2−4가 완전제곱수(perfect square)인 것이다.