A computational mathematics workbook for studying d(n) = n − φ(n) − τ(n), where φ is Euler's totient and τ is the divisor-counting function.
An Excel-based research instrument where every claim is computed by Excel formulas, not pre-stated. Charts discover structure. Conditional formatting flags anomalies. If a theorem fails, the spreadsheet shows it.
Deficiency_Research_Instrument.xlsx — 8 sheets, ~3000 live formulas:
| Sheet | What it computes | Key result |
|---|---|---|
| Computation Engine | d(n) for n=1..1000 via =A-B-C, τ(n) via SUMPRODUCT |
1000 rows, all predictions verified |
| Zero Set Proof | Exhaustive case analysis + 100K scan | d(n)=0 iff n ∈ {6,8,9} |
| Consecutive Coincidences | All pairs where d(n)=d(n+1) up to 100K | Only 7 pairs, all Fermat-related |
| Fiber Concentration | Top 200 fiber sizes, mod 30 analysis | Hardy-Littlewood concentration at k≡25(30) |
| Ray Structure | Multi-series scatter: primes/semiprimes/p²/composites | Visible ray separation by number type |
| Phase Transition | r(n) = (n−φ(n))/τ(n) for n=2..500 | r(n)=1/2 iff n is prime — sharp floor |
| GCD-Projection | cos²(θ*)=1/φ bridge identity + 200 GCD homomorphism tests | Identity proven to machine precision |
| Parity Theorem | d(n) ≡ n (mod 2) iff n is not a perfect square | 498 values verified |
- Zero Set: d(n) = 0 ⟺ n ∈ {6, 8, 9}
- Spectrum Gap: d(n) ≠ 1 for all n
- Prime Formula: d(p) = −1 for all primes p
- Prime Power: d(p^k) = p^(k−1) − k − 1
- Semiprime: d(pq) = p + q − 5 for distinct primes p < q
- Isospectral Identity: d(2p) = d(p²) = p − 3 for all odd primes p
- Phase Transition: The parametric zero set Z(α) transitions sharply at α = 1/2
- Fiber Finiteness: All fibers F_k are finite for k ≥ 0
- Parity: d(n) ≡ n (mod 2) ⟺ n is not a perfect square
- Fermat Coincidence: d(2^(2^m)−1) = d(2^(2^m)) for m = 1,2,3,4
- Squarefree Formula: d(p₁⋯pₖ) = ∏pᵢ − ∏(pᵢ−1) − 2^k
- Average Order: (1/N)∑ d(n)/n → 1 − 6/π² ≈ 0.3921
- Dirichlet Series: ∑ d(n)/n^s = ζ(s−1) − ζ(s−1)/ζ(s) − ζ(s)²
- GCD Bridge Identity: cos²(θ*) = 1/φ where θ* = arccos(1/√φ)
- Modular Reduction: gcd(ab, n) = gcd(gcd(a,n)·gcd(b,n), n) for all positive a,b,n
Every claim in this workbook passes one test: can it stand under its own mathematical justification?
- If it's a theorem, it has a proof
- If it's a computation, Excel verifies it independently
- If it's a conjecture, it's clearly marked as such
- If it can be falsified, the workbook tries
The Python scripts (develop_v2.py etc.) generate the Excel workbooks using openpyxl. The workbooks contain live Excel formulas — they are not static reports.
pip install openpyxl
python develop_v2.py- Python 3.10+ with openpyxl
- Microsoft Excel (for viewing charts and live formulas)