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 $m + n \ge d$

**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 $1 \le j \le n$ with $\gcd(A_1,...,A_n, B_1,...B_n) = 1$

Let $d=min(h(A_1), h(A_2), ...., h(A_n), h(B_1),...,h(B_m)$.

>> **Conjecture:** if $\sum_{i=1}^{n} A_i = \sum_{j=1}^{m} B_j$ then $m + n \ge d$

**See also:**

* [Lander, Parkin, and Selfridge conjecture](https://en.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_conjecture)

* [Minimal exponent in prime factorization of n A051904](https://oeis.org/A051904)

* [Nivens Constant](http://mathworld.wolfram.com/NivensConstant.html)