In this Note, We propose a new definition called "large radical of an integer". Using this definition, three very useful $AB$ conjecture are given. In particular, every theorem-conjecture that is the consequence of the $ABC$ conjecture is also a consequence of the first $AB$ conjecture. But the true possibility of the first $AB$ conjecture is higher. As an electrical engineer, I am not able to deeply study on the conjectures. Hopefully, the conjectures would be interested by a professor in the field and be researched further.

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 $B$.

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

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.

The conjecture was proposed by Dao Thanh Oai, Thai Binh, Viet Nam.