Super Golden Ratio

[Albert Chan]

(08-09-2022 10:45 AM)Gerson W. Barbosa Wrote:  300766 ENTER 33 1/X Y↑X is shorter for 12 digits and the approximation is slightly better.

Let (α,β,γ) be roots of x^3 = x^2 + 1
We can build cubic with roots (-α^n, -β^n, -γ^n)

n=1: x^3 + x^2 + 1
n=2: x^3 + x^2 - 2x + 1
n=3: x^3 + 4x^2 + 3x + 1

x^2 term coefficient = (α^+n + β^+n + γ^+n)
x^1 term coefficient = (α^−n + β^−n + γ^−n), since αβγ = 1

x^+n = x^+(n-3) + x^+(n-1)
x^−n = x^−(n-3) − x^−(n-2)

lua> B, C = {1,1,4}, {0,-2,3}
lua> for n=4,40 do B[n]=B[n-3]+B[n-1]; C[n]=C[n-3]-C[n-2] end
lua> for n=30,36 do print(n, B[n], C[n]) end

Code:

30      95545   406
31      140028  403
32      205221  -870
33      300766  3
34      440794  1273
35      646015  -873
36      946781  -1270

For n=33, linear term coefficient is relatively tiny. Also, (α^n, β^n, γ^n) well separated.

lua> surd(B[33], 33) -- estimated Psi
1.4655712318782408

--

lua> B[33]^2-2*C[33], C[33]^2-2*B[33] --Graeffe Root Squaring
90460186750      -601523

Free42: (90460186750)^(1/66)
1.465571231876768025024430123337594

Free42: (90460186750^2 + 2*601523)^(1/132) // "square" again
1.465571231876768026656731225219939