My question: Are the conjectures as follows correct?
Given a positive integer $P>1$, let its prime factorization be written
$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_k^{a_k}$$.
Define the functions $h(P)$ by $h(1)=1$ and $h(P)=min(a_1,a_2,...,a_k)$
Case 1: Let $n \ge 1 $ be positive integers, and $A_i \ne B_j$ are positive integers for all $1 \le i \le n$ and $1 \le j \le n$ with $\gcd(A_1,...,A_n, B_1,...B_n) = 1$
Let $d=min(h(A_1), h(A_2), ...., h(A_n), h(B_1),...,h(B_n))$.
Conjecture: if $\sum_{i=1}^{n} A_i = \sum_{j=1}^{n} B_j$ then $2n \ge d$
Case 2: Let $n \ne m$ and $n, m \ge 1 $ be positive integers, and $A_i, B_j$ are positive integers for all $1 \le i \le n$ and $1 \le j \le m$ with $\gcd(A_1,...,A_n, B_1,...B_m) = 1$
Let $d=min(h(A_1), h(A_2), ...., h(A_n), h(B_1),...,h(B_m))$.
Conjecture: if $\sum_{i=1}^{n} A_i = \sum_{j=1}^{m} B_j$ then $m + n \ge d$
See also: