este o ecuatie teoretica :
x² + x +1 = 0 nu are solutia 1
se inmulteste cu ( x - 1)
( x -1)· ( x² + x +1 ) = 0
x³ -1 = 0 ⇒ deci oricare solutie x₁ , x₂
avem x₁³ =1 ; x₂³ =1
puteri :
daca n = 3· k ( multiplu de 3)
x₁ ( la n ) + 1 / x₁ ( la n) = 1 + 1 = 2
n = 3· k + 1 , cu propr. puteri
x₁¹ + 1 / x₁¹ = ( x₁² + 1) / x₁ = ( - x₁) / x₁ = - 1
radacina x₁ , verifica ecuatia x₁² + x₁ + 1 =0 ; x₁² + 1 = - x₁
n = 3· k + 2 , cu propr. puteri
x₁² + 1 / x₁² = ( x₁⁴ + 1) / x₁² = ( x₁³x₁ + 1) / x₁² = ( x₁ + 1) / x₁² =
dar x₁² = - x₁ - 1
= ( x₁ + 1) / ( -x₁ - 1) = - 1
