The abc Conjecture Connection

Statement of abc

abc Conjecture. For every $\varepsilon > 0$, there exists $K(\varepsilon)$ such that for all coprime positive integers with $a + b = c$:

$$c < K(\varepsilon) \cdot \text{rad}(abc)^{1+\varepsilon}$$

where $\text{rad}(n)$ is the product of the distinct prime factors of $n$.

The Minimal Radical

The Collatz map involves only two primes: 2 and 3. The radical of any product of their powers is:

$$\text{rad}(2^a \cdot 3^b) = 6$$

regardless of the exponents. This is the smallest possible radical for a product of two distinct primes. The abc conjecture is maximally constraining precisely in this setting — the terms can be enormous ($2^{485}$, $3^{306}$), yet the radical stays at 6.

Application to Collatz Cycles

The cycle equation from the Affine Orbit Structure:

$$3^S \cdot n + C_\text{total} = 2^E \cdot n$$

can be rewritten as:

$$(2^E - 3^S) \cdot n = C_\text{total}$$

Setting $a = 3^S \cdot n$, $b = C_\text{total}$, $c = 2^E \cdot n$:

$$\text{rad}(abc) = \text{rad}(2^E \cdot 3^S \cdot n^2 \cdot C_\text{total}) \leq 6 \cdot n \cdot C_\text{total}$$

The abc conjecture says $c < K(\varepsilon) \cdot \text{rad}(abc)^{1+\varepsilon}$, giving:

$$2^E \cdot n < K(\varepsilon) \cdot (6 \cdot n \cdot C_\text{total})^{1+\varepsilon}$$

For large cycles ($E$ large), this forces $n$ to be bounded — you can't have arbitrarily large cycles of a given pattern.

Hierarchy of Bounds

Method	Bound on $\beta(s)$	Bound on cycle $n$	Status
Irrationality of $\log_2 3$	$\beta > 0$	Finite for each pattern	Proved
Roth's theorem	$\beta > c/s$	$n < \exp(c \cdot s^2)$	Proved
Baker's theorem	$\beta > \exp(-c \cdot \log s \cdot \log\log s)$	$n < \exp(\exp(c \cdot s))$	Proved
Pillai conjecture	$\beta > c$ (constant)	Much tighter	Unproved
abc conjecture	$\beta > 2^{-\varepsilon k}$	Exponential bound on $n$	Unproved

Each row strengthens the previous. The abc conjecture gives the strongest constraints but remains unproved. Our divisibility obstruction is an independent constraint from the affine structure, orthogonal to all of these.

S-Unit Equations

The equation $|2^E - 3^S| = g$ is a special case of the S-unit equation with $S = \{2, 3\}$.

Evertse (1984) proved: for each fixed $g$, there are finitely many solutions $(E, S)$. This means:

• Each gap value $g$ can only appear for finitely many convergents
• The gaps $|2^E - 3^S|$ grow without bound along the convergent sequence

Combined with our framework: if the divisibility obstruction holds for all gaps up to some bound $G$, and Baker's theorem shows all convergents beyond a certain size have gap $> G$, then all cycles are eliminated by a finite computation.

The Punchline

The Collatz conjecture lives in the gap between what we can prove about powers of 2 and 3, and what the abc conjecture says must be true. The two smallest primes, the smallest radical, the tightest constraint. Collatz is the atomic case of abc.

Related

• Convergent Elimination — the computational elimination that abc would strengthen
• Divisibility Obstruction — our independent algebraic constraint
• Bit Destruction Bound — where Roth's theorem and Pillai enter
• Path to Proof — how all pieces fit together