Given a positive integer $P>1$, let its prime factorization be written as$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}.$$

Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k).$

 Is new the $n$-conjecture, formulated as follows, correct?

**Conjecture:** if ${P_1,P_2,...,P_n}$ are positive integer and pairwise coprime, then,

 $$\min\{h(P_1), h(P_2),...,h(P_n), h(P_1+P_2+...+P_n)\} \leq n+1.$$

I proposed the case $n=2$ two years ago here ([Is the conjecture A+B=C following correct?](https://mathoverflow.net/questions/303141/is-the-conjecture-ab-c-following-correct)). Now I reformulate that question as follows:

> Let ${P_1,P_2}$ are coprime, then:
> $$\min\{h(P_1), h(P_2), h(P_1+P_2)\} \leq 3$$