Connections to Classical Mathematics

The structural results on this site connect to deep areas of number theory. The Collatz conjecture sits at the intersection of 2-adic analysis, Diophantine approximation, and the abc conjecture.

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• abc Conjecture — Constrains how close powers of 2 and 3 can be. The Collatz map involves only primes 2 and 3, making it the minimal-radical case. Connects to: cycle elimination, bit destruction.

• Roth's Theorem — Bounds the irrationality measure of $\log_2 3$, giving $\beta(s) > c/s$. Connects to: Bit Destruction Bound.

• Baker's Theorem — Effective lower bounds on linear forms in logarithms: $|k \log 2 - s \log 3| > \exp(-c \cdot \log k \cdot \log \log k)$. Gives the strongest unconditional bound on cycle length. Connects to: Convergent Elimination.

• Pillai's Conjecture — $|2^m - 3^n| \to \infty$ as $m + n \to \infty$. Would give $\beta(s) > c$ (constant), upgrading convergence to $O(\log n)$ drops. Connects to: Bit Destruction Bound.

• S-Unit Equations (Evertse, 1984) — The equation $2^a - 3^b = g$ has finitely many solutions for each fixed $g$. Connects to: Divisibility Obstruction.

• Terras's Theorem (1976) — For almost all $n$ (density 1), the Collatz orbit eventually drops below $n$. Our mixing results give a cleaner proof. Connects to: 3-Adic Mixing.

• Weyl Equidistribution — The sequence $\{s \cdot \log_2 3\}$ is equidistributed mod 1, meaning slow and fast sets are interleaved without pattern. Connects to: Bit Destruction Bound.