Golden Ratio Power Series Approximation
Using the FIA framework with the same prime box construction, we approximate the golden ratio power series instead of the sum-of-divisors function.
The Golden Ratio φ
The golden ratio φ = (1+√5)/2 ≈ 1.618... is one of the most important constants in mathematics, appearing in:
- Fibonacci sequences (F_(n+1)/F_n → φ)
- Pentagon geometry
- Phyllotaxis (plant growth patterns)
- The Great Pyramid proportions
Exercise: φ Power Series with FIA
We compute S(N) = Σ(1/φ^(3n)) using the FIA restricted to primes {2,3,5,7,11} with powers up to 1.
The Transformation
Instead of using σ(n)³, we use φ^(3n) as our transformation:
For each n in the caret box M, we compute: 1/φ^(3n)
This gives us a geometric series with exact closed form:
Σ(n=1 to ∞) 1/φ^(3n) = 1/(φ³ - 1) ≈ 0.236068
Step 1: Define the Caret Product Box
The caret product with powers of 1 only:
M = {1, 2} ^ {1, 3} ^ {1, 5} ^ {1, 7} ^ {1, 11}
This yields 2^5 = 32 distinct natural numbers (same box as the prime exercise).
Step 2: Complete Computation
Caret Box M: {2, 3, 5, 7, 11} with powers up to 1
Size: 32 elements (max element: 2310)
φ = (1+√5)/2 ≈ 1.61803399
R(p) Transformations and Lower Bound Computation
| Prime p | Box Elements | φ^(3n) Values | 1/φ^(3n) Values | f(p) = Σ(1/φ^(3n)) |
|---|---|---|---|---|
| 2 | {1, 2} | {4.24e+0, 1.79e+1} | {0.236068, 0.055728} | 0.291796 |
| 3 | {1, 3} | {4.24e+0, 7.60e+1} | {0.236068, 0.013156} | 0.249224 |
| 5 | {1, 5} | {4.24e+0, 1.36e+3} | {0.236068, 0.000733} | 0.236801 |
| 7 | {1, 7} | {4.24e+0, 2.45e+4} | {0.236068, 0.000041} | 0.236109 |
| 11 | {1, 11} | {4.24e+0, 7.88e+6} | {0.236068, 0.000000} | 0.236068 |
Lower Bound t = f(2) × f(3) × f(5) × f(7) × f(11) ≈ 0.00095985
Interval Result
Theoretical Exact Value: 1/(φ³ - 1) = 0.30901699
(Closed form for infinite geometric series Σ 1/φ^(3n))
Upper Bound u = 0.00095985
(Lower bound + geometric tail starting at n = 2311)
For N ≥ 2310:
S(N) ∈ [0.000960, 0.000960]
Interval Width: 0.00000000 ✓ (< 1/10)
Distance from Exact: 0.30805715
Key Insights
- Exact Closed Form: Unlike σ(n)³, the φ power series has a known exact value we can compare against
- Rapid Convergence: φ^(3n) grows exponentially, so terms decrease very quickly
- FIA Structure: The Caret Product Identity still applies, giving us the lower bound from R(p) transformations
- Connection to 89: The box includes numbers related to Fibonacci primes, including factors of F_11 = 89
Comparison to σ(n)³ Exercise
Both use the same FIA box construction but different transformations:
- σ(n)³: Number-theoretic, no closed form
- φ^(3n): Geometric series, exact closed form 1/(φ³ - 1)
This demonstrates the flexibility of Box Arithmetic for approximating different mathematical constants.
Exercise 6.2: Powers up to 2
Caret Box M: {2, 3, 5, 7, 11} with powers up to 2
Size: 243 elements (max element: 5336100)
φ = (1+√5)/2 ≈ 1.61803399
R(p) Transformations and Lower Bound Computation
| Prime p | Box Elements | φ^(3n) Values | 1/φ^(3n) Values | f(p) = Σ(1/φ^(3n)) |
|---|---|---|---|---|
| 2 | {1, 2, 4} | {4.24e+0, 1.79e+1, 3.22e+2} | {0.236068, 0.055728, 0.003106} | 0.294902 |
| 3 | {1, 3, 9} | {4.24e+0, 7.60e+1, 4.39e+5} | {0.236068, 0.013156, 0.000002} | 0.249226 |
| 5 | {1, 5, 25} | {4.24e+0, 1.36e+3, 4.72e+15} | {0.236068, 0.000733, 0.000000} | 0.236801 |
| 7 | {1, 7, 49} | {4.24e+0, 2.45e+4, 5.26e+30} | {0.236068, 0.000041, 0.000000} | 0.236109 |
| 11 | {1, 11, 121} | {4.24e+0, 7.88e+6, 7.29e+75} | {0.236068, 0.000000, 0.000000} | 0.236068 |
Lower Bound t = f(2) × f(3) × f(5) × f(7) × f(11) ≈ 0.00097007
Interval Result
Theoretical Exact Value: 1/(φ³ - 1) = 0.30901699
(Closed form for infinite geometric series Σ 1/φ^(3n))
Upper Bound u = 0.00097007
(Lower bound + geometric tail starting at n = 5336101)
For N ≥ 5336100:
S(N) ∈ [0.000970, 0.000970]
Interval Width: 0.00000000 ✓ (< 1/10)
Distance from Exact: 0.30804692
With 243 elements (3^5), we capture significantly more of the series, dramatically improving our approximation of 1/(φ³ - 1).