Interactive visualization of the three-way classification: ntrim | ctrim | conjugate
© Lee McCulloch James
The parametrisation (m, n) → (m² − n², n(2m−n), m² − mn + n²) induces a three-way partition:
m + n ≡ 0 (mod 3)
gcd(a,b,c) = 3
m + n ≢ 0, m > 2n
gcd(a,b,c) = 1, a > b
m + n ≢ 0, m < 2n
gcd(a,b,c) = 1, a < b
The mod-3 partition of {(m,n) : gcd(m,n) = 1, m > n > 0}
Click a point in the parameter space to see details
| m | n | a | b | c | m+n mod 3 | m : 2n | gcd(a,b,c) | Lineage |
|---|
Verification that odd generators (both types) square to Odd-1 outputs, while even generators remain even.
| m | n | m+n mod 3 | Generator Parity | a | b | a+b mod 3 | Output Parity |
|---|
Each canonical primitive (a, b, c) has a conjugate (b, a, c) arising from the parameter transformation (m, n) ↔ (m, m−n).
| Canonical (m > 2n) | c | Conjugate (m < 2n) | ||||
|---|---|---|---|---|---|---|
| (m, n) | a | b | b | a | (m, m−n) | |