## Need to confirm if this is right

For discussing Olympiad Level Number Theory problems
Asif Hossain
Posts: 171
Joined: Sat Jan 02, 2021 9:28 pm

### Need to confirm if this is right

Problem: Find all positive integers $x,y$ such that $p^x-y^p=1$ where $p$ is prime.(Czech Slovakia 1996)
Solution:(Need to confirm )
Hmm..Hammer...Treat everything as nail

~Aurn0b~
Posts: 44
Joined: Thu Dec 03, 2020 8:30 pm

### Re: Need to confirm if this is right

Asif Hossain wrote:
Thu Apr 22, 2021 12:56 pm
Problem: Find all positive integers $x,y$ such that $p^x-y^p=1$ where $p$ is prime.(Czech Slovakia 1996)
Solution:(Need to confirm )
I think its correct, however, the solution could be more short, after figuring out, $1+y=p^{x-1}$(Assuming p is odd ofc) , we can say that $\frac{y^p+1}{y+1}=p\Rightarrow 1+y \geq \frac{y^p+1}{y+1} \Rightarrow y^2+2y\geq y^p \Rightarrow y+2\geq y^{p-1}\Rightarrow 2\geq y(y^{p-2}-1)$

Asif Hossain
Posts: 171
Joined: Sat Jan 02, 2021 9:28 pm

### Re: Need to confirm if this is right

Nice Solution the idea of bounding didn't come to my mind. mine was rather unnecessarily long
Hmm..Hammer...Treat everything as nail