Every ring in the hierarchy has coupling classes: elements grouped by which primes divide them. Each prime is present or absent in gcd(n,N), giving 2^k classes for k primes. Z/2,310 (5 primes): 32 classes. Z/12,612,600 (6 primes): 64 classes. Z/214,414,200 (7 primes): 128 classes. The class count doubles with each new prime.
| Ring | N | Primes | Classes |
|---|---|---|---|
| Z/210 | 2*3*5*7 | 4 | 16 = 2^4 |
| Z/2,310 | 2*3*5*7*11 | 5 | 32 = 2^5 |
| Z/970,200 | 2^3 * 3^2 * 5^2 * 7^2 * 11 | 5 (prime-power) | 32 = 2^5 |
| Z/12,612,600 | Z/970,200 * 13 | 6 | 64 = 2^6 |
| Z/214,414,200 | Z/12,612,600 * 17 | 7 | 128 = 2^7 |
Z/970,200 has the same 5 primes as Z/2,310 but with raised exponents (higher powers). Same 32 class patterns, more elements per class. Z/12,612,600 adds 13: classes double to 64. Z/214,414,200 adds 17: classes double again to 128. Each new prime doubles the anatomy.
| Class | Count | Coupling | Divisor |
|---|---|---|---|
| Units (gcd=1) | 480 | 2310 (max) | coprime to all |
| 2-pure (gcd=2) | 480 | 1155 | divisible by 2 only |
| 3-pure (gcd=3) | 240 | 770 | divisible by 3 only |
| 5-pure (gcd=5) | 120 | 462 | divisible by 5 only |
| 7-pure (gcd=7) | 80 | 330 | divisible by 7 only |
| 11-pure (gcd=11) | 48 | 210 | divisible by 11 only |
| Void (0) | 1 | 0 | all primes divide |
In Z/214,414,200 (128 classes), two more pure classes appear: 13-pure and 17-pure. 1 unit + 7 pure + 119 compounds + 1 void = 128 = 2^7.
Enter any number. See its class, coupling, and all 7 channel statuses in Z/214,414,200. Units have maximum coupling (38,707,200 = phi(214,414,200)). Void has zero.
Analyze any element:
Try: 1 (unit), 2 (2-class), 13 (13-class), 17 (17-class), 6 (compound), 0 (void).
Seven primes create eight coupling classes: the unit class plus one pure class per prime. In Z/214,414,200, all eight are present. In smaller rings, only the primes that divide N appear.
The coupling heatmap reveals different structure depending on the coloring:
Each ring adds structure. Z/2,310 is the pedagogical entry (squarefree). Z/970,200 raises exponents to prime powers. Z/12,612,600 adds 13. Z/214,414,200 adds 17. The architecture scales:
| Ring | N | Classes | phi(N)/N |
|---|---|---|---|
| Z/210 | 210 | 16 = 2^4 | 8/35 |
| Z/2,310 | 2310 | 32 = 2^5 | 16/77 |
| Z/970,200 | 970200 | 32 = 2^5 | 16/77 |
| Z/12,612,600 | 12612600 | 64 = 2^6 | 192/1001 |
| Z/214,414,200 | 214414200 | 128 = 2^7 | 3072/17017 |
Class count doubles with each new prime (binary choice). Element count scales by Carmichael lambda = 420 from Z/2,310 to Z/970,200 (raising exponents preserves class structure). 13 and 17 each double the anatomy without raising exponents.
| Aspect | Standard View | Ring Structure View |
|---|---|---|
| 128 classes | Divisor lattice of 214,414,200 | 2^7 = seven binary channel choices. Each ring doubles. |
| 8 pure classes | Divisor subgroups | Unit class plus one pure class per prime. Structured hierarchy. |
| Coupling strength | Coprimality measure | coupling(n) = N/gcd(n,N). Measures how connected an element is. |
| Ring hierarchy | Different moduli | Z/2,310 -> Z/970,200 -> Z/12,612,600 -> Z/214,414,200. Each adds structure. |
| Heatmap | Visualization technique | Four independent colorings of the same ring |
| phi(N)/N | Euler product formula | Fraction of units tightens: 20.78% (Z/2,310) -> 18.05% (Z/214,414,200) |
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