[In this Note](https://drive.google.com/file/d/1O7mPNseiyABmD1Sbra1l8CWuxV9udX9i/view?usp=sharing), We propose a new definition called *"large radical of an integer"*. Using this definition, three very useful $AB$ conjecture are given.

**1. Large counter examples of the ABC conjecture** 

By [Fundamental theorem of arithmetic](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/w/index.php?title=Abc_conjecture&section=5#Computational_results), there are $14482065$ case $A+B > rad(AB(A+B))$ in ranges $1 \le A < B <10^8$. 

Maybe we can prove that in the most counter example of $A+B < rad(AB(A+B)$ has at least one exponents of $A$ or $B$ or $(A+B)$ greater than $2$ (the exponents of $A$ or $B$, $A+B$ greater than $2$ refer at least one $x_i$ or $y_j$ or $z_k$ $>2$). But in the range $1 \le A < B <10^8$, there are maximum  $\approx 508544975$ numbers have an exponents greater than $2$. So in range $1 \le A < B <10^8$ there are $14482065$ counter examples, this is **LARGE COUNTER EXAMPLES.**

**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 $rad(AB(A+B)) < A+B< rad'(A).rad'(B).rad'(A+B)$

[![enter image description here][3]][3]

[Other two conjecture in here](https://drive.google.com/file/d/1O7mPNseiyABmD1Sbra1l8CWuxV9udX9i/view?usp=sharing)

COMPUTER CHECKED

[![enter image description here][4]][4]


>> **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.


  [1]: https://i.sstatic.net/RpbI1.png
  [2]: https://i.sstatic.net/ykNgR.png
  [3]: https://i.sstatic.net/Zb7F2.png
  [4]: https://i.sstatic.net/NnULv.png