Could You give your ideas, your comment, or a referen for a conjecture as follows:

Consider $A, B, C$ be three positive integers numbers. By Fundamental theorem of arithmetic we write:

$A=a_1^{x_1}a_2^{x_2}...a_n^{x_n}$,

$B=b_1^{y_1}b_2^{y_2}...b_m^{y_m}$,

$C=c_1^{z_1}c_2^{z_2}...c_k^{z_k}$

Let $N=\min\{x_i, y_j, z_h \}$ where $1 \le i \le n, 1\ \le j \le m, 1\le h \le k$

For a positive integer $N_0 > 3$, there exist only finitely many triples $(A, B, C)$ of coprime positive integers, with $A+B=C$, such that: $N \ge N_0$

**Edited:** I repalced $N_0 \ge 3$ by $N_0 > 3$