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> $\DeclareMathOperator\rad{rad}$**Conjecture:** *If $A, B, C$ are positive integers with $\gcd(A, B)=1$, $\gcd(B, C)=1$, and $\gcd(C, A)=1$, and if $A+B=C$, then $\min(A,B) \le \rad(ABC)$.*
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If the conjecture is valid, then we can use the conjecture to prove the Fermat last theorem as follows:


**Proof of the Fermat last theorem:**


We consider the Fermat equation: 


$x^n+y^n=z^n$ with $\gcd(x, y)=1$, $\gcd(y, z)=1$, and $\gcd(z, x)=1$.


There is no loss of generality in assuming that $x \le y <z$. By the conjecture, we get $x^n \le \rad(x^n y^n z^n)=\rad(x y z)\le x y z $.


So $x^n+y^n \le xyz+y^n<z^3+(z-1)^n$.


But we can easily prove that $z^3+(z-1)^n < z^n$ whenever $n > 3$ and $z>1$. So now we only need to prove the Fermat last theorem with $n=3$.



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> **My question:** Is the conjecture above new and correct?
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Is the conjecture $min(A,B) \le rad(ABC)$ new and correct?
Note,
$$625+2048=5^4 + 2^{11} = 3^5\times 11 = 2673.$$
The relevant radical is thus $$2\times 3 \times 5 \times 11=330.$$
Therefore, the conjecture is false.


