Let $a, b, c$ be coprime positive integers such that $a+b=c$. Define $h(n)$ as the minimum exponent in the prime factorization of $n$ (with the convention $h(1) = +\infty$ and $1/h(1) = 0$).

Conjecture: $$\frac{1}{h(a)} + \frac{1}{h(b)} + \frac{1}{h(c)} \ge 1 - \frac{C^*}{\ln(\ln c)}$$

Through extensive computational surveying, the critical constant $C^*$ is found to be maximized at the triple $(1, 12167, 12168)$, where $12167 = 23^3$ and $12168 = 2^3 \cdot 3^2 \cdot 13^2$: $$C^* = \frac{1}{6} \ln(\ln 12168) \approx 0.373567972$$

The Upper Bound Corollary: If we define $S = \frac{1}{h(a)}+\frac{1}{h(b)}+\frac{1}{h(c)}$, then for any triple with $S < 1$: $$c \le \exp\left(\exp\left(\frac{C^*}{1 - S}\right)\right)$$

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