In this Note, We propose a new definition called "large radical of an integer". Using this definition, three very useful $AB$ conjecture are given.

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 $h(P)$, $d(P)$ and $rad'(P)$ by $h(1)=1$ and $h(P)=min(a_1, a_2,..,a_k)$

Let $g=\gcd(a_1, a_2,..., a_k)$

$$d(P)=\frac{h(P)}{g}=min(\frac{a_1}{g},\frac{a_2}{g},...,\frac{a_k}{g})$$ $$rad'(P)=(p_1p_2...p_k)^{d(P)}$$

Some examples:

1. Let $P=2^5.5^7.11^8$ then $rad(P)=2.5.11$ and $rad'(P)=2^5.5^5.11^5$

2. $P=17^8$ then $rad(P)=17$ and $rad'(P)=17$

There are some simple properties of $rad'(P)$

1. $rad'(P)=rad(P)^{d(P)}$

2. $rad(P) \le rad'(P) \le P$

3. $rad'(P^n)=rad'(P) \le P$

4. In general case $rad'(AB) \ne rad'(A)rad'(B)$

Conjecture1: For every positive real number $\varepsilon >0$, the inequality $$A+B > (rad'(A).rad'(B).rad'(A+B))^{1+\varepsilon}$$ has only finitely relatively prime integers $A$ and $B$.

Remarks: By the definition, the first AB conjecture is weaker than the ABC conjecture. The proof of first AB is simpler than the proof of ABC conjecture because $rad'(A) \ge rad(A)$. The true possibility of the first AB conjecture is higher than the ABC conjecture. But the first AB conjecture is as useful as the ABC conjecture. Why? Because useful ABC conjecture based on two peroperties $rad(A^n)=rad(A)$ and $rad(A) \le A$. The first AB conjecture also have two properties $rad'(A^n)=rad'(A)$ and $rad'(A) \le A$.

Let $\varepsilon=0$ here are some examples $rad(AB(A+B)) < A+B< rad'(A).rad'(B).rad'(A+B)$

Other two conjecture in here

COMPUTER CHECKED

My question: Could You help me full fill the table 2 above? Hopefully, the conjectures would be interested by a professor in the field and be researched further.