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

1. Large counter example of the ABC conjecture

By Fundamental theorem of arithmetic

$A=a_1^{x_1}a_2^{x_2}...a_n^{x_n}$, $B=b_1^{y_1}b_2^{y_2}...b_m^{y_m}$, $C=c_1^{z_1}c_2^{z_2}...c_k^{z_k}$.

The ABC conjecture states that:

ABC Conjecture. For every positive real number $\varepsilon$, there exist only finitely many triples $(A, B)$ of coprime positive integers, such that: $A + B > \operatorname{rad}(abc)^{1+\varepsilon} $.

If the $ABC$ conjecture is true, it must true with $\varepsilon=0$.

In this table, there are $14482065$ case $A+B > rad(AB(A+B))$ in ranges $1 \le A < B <10^8$.

Maybe we can prove that in most the counter example of $A+B < rad(AB(A+B)$ has at least one number $A$ or $B$ or $(A+B)$ with exponents $x_i, y_i, z_i >2$. But in the range $1 \le A < B <10^8$ we have maximum $1037611$ numbers have an exponents greater than $2$. So in range $1 \le A < B <10^8$ there are $14482065$ counter example, this is large counter example.

II. The first AB conjecture

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. The true possibility of the first AB conjecture is higher than the ABC conjecture because $rad'(A) \ge rad(A)$. But the first AB conjecture is as useful as the ABC conjecture. 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? OR Hopefully, the conjectures would be interested by a professor in the field and be researched further.