Among $168$ prime numbers in range $1$ to $10^3$, there are $84$ prime numbers $n$ such that: $p^k> n.rad(p^{k+1}−n)$ where $1 \le n<p$ and $k=2,3,4$. There are also $84$ prime numbers $n$ such that: $p^k> (n.rad(p^{k+1}−n))^{1.01}$ where $1 \le n<p$ and $k=2,3,4$. The ratio $\frac{84}{168}=0.5$ is high.

**Question 1:** Are there infinitely many primes $p$, positive integers $k, n$ such that $1 \le n < p$ and $p^k > n.rad(p^{k+1}−n)$?

**Question 2:** If the answer to question 1 is yes, is there $\varepsilon > 0$ such that there are infinitely many primes $p$, positive integers $k, n$ such that $1 \le n < p$ and $p^k > (n.rad(p^{k+1}−n))^{1+\varepsilon}$?

**Question 3:** If the answer of question 2 is yes, then is the ABC conjecture false? if we let $A=p^{k+1}-n$, $B=n$

$A+B=p^{k+1}-n+n=p^{k+1}=p.p^k>p.(n.rad(p^{k+1}-n))^{1+\varepsilon} \ge (p.n.rad(p^{k+1}-n))^{1+\varepsilon_0}=rad(ABC)^{1+\epsilon_0}$

List of $84$ pair $(p, n)$ as follows $p=A(2i-1)$, $n=A(2i)$ for $i=1,2,...,84$ 

Examples:

$13^2-10.rad(13^3-10)=139>0$

$23^2-17.rad(23^3-17)=19>0$

$37^2-28.rad(37^3-28)=949>0$


$A=$ [13    10    23    17    37    28    73    55   107    43   137    89   181   136   191 89   281    41   313     1   337   253   353    89   379     3   383   287   433    17 433   325   467     5   541   406   563   422   631    31   769   577   811    83   863 647   937   703     3     1     5     1     7     1    17     1    19     1    31     1 41     1    53     1    73    46    97     1   107     1   127     1   127    19   131 121   163     1   181    49   193     1   199     1   239     1   241     1   251     1 257     1   271     1   307     1   331    49   337     1   443     1   449     1   487 1   557     1   577     1   593     1   599   193   751     1   797    49   821   561 881   385   907   817   937   289   983   289     3     1    23     2    29     5    43 15    47     7   103     7   163    82   197    17   211     3   229    24   251     1 277     8   281   249   283    43   449   193   487     1   563    11   821     5   827 11   853     5]