+129771 a^3 - 1/a^3
a^2 - a -1 = 0
a^2 -a + 1/4 = 1 + 1/4
(a - 1/2)^2 = 5/4 take both roots
a -1/2 = sqrt 5/2 or a -1 = -sqrt 5/2
a = [1 + sqrt 5]/2 a = [ 1 -sqrt 5]/2
a^3 = [ 1 +sqrt 5] /2 * [1 + sqrt 5]/2 *[1 +sqrt 5] /2
a^3 = [ 1 + 2sqrt 5 + 5 ] / 4 * [ 1 + sqrt 5 ] /2
a^3 = [ 1 + 2sqrt 5 + 5 + sqrt 5 + 10 + 5sqrt 5] / 8
a^3 = [16 + 8sqrt 5] / 8 = 2 + sqrt 5
1/a^3 = 1/ [2 + sqrt 5] = [2-sqrt 5] / [4-5] = sqrt 5 -2
So in one case a^3 - 1/a^3 = [2 + sqrt 5] -[ sqrt 5 - 2] = 4
In the other case
a^3 = [ 1 -sqrt 5]/2 * [1 -sqrt 5]/2 * [1 -sqrt 5]/2
a^3 = [ 1 - 2sqrt 5 + 5]/4 * [ 1 -sqrt 5]/2
a^3 = [ 1 -2sqrt 5 + 5 - sqrt 5 +10 -5sqrt 5] /8
a^3 = [ 16 - 8sqrt 5]/8 = 2 -sqrt 5
1/a^3 = 1/[2 - sqrt 5] = [2 + sqrt 5] / [4-5] = -2-sqrt 5
So in the other case a^3 - 1/a^3 = [2-sqrt 5 ] - [- 2 - sqrt 5 ] = 4 (same result !!!)
