Find all prime numbers $p$ and integers $a$ and $b$ (not necessarily positive) such that $p^a + p^b$
is the square of a rational number.
BdMO 2010 H. Sec. problem 8
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
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Please install LaTeX fonts in your PC for better looking equations,
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Re: BdMO 2010 H. Sec. problem 8
Hint: first think about when $p^a+p^b=n^2$, for some $n\in\mathbb N$.
"Everything should be made as simple as possible, but not simpler." - Albert Einstein
Re: BdMO 2010 H. Sec. problem 8
I am posting to find all such primes $p$. Let wlog $a\ge b$ (even if both negative or one positivd another negative). If $a=b,p=2$.Now let $a=b+c,c>0$.So $p^b(p^c+1)$ is a rational square.Since $gcd(p^b,p^c+1)=1$ when $b>0$or $gcd(p^d,p^c+1)=1$ when $b=-d,d>0$; we see that $p^c+1$ is a square. Now let $p^c+1=k^2 \Longrightarrow p^c=(k-1)(k+1)$. Then $k+1=p^x,k-1=p^y;p^x-p^y=p^y(p^{x-y}-1)=2 \Longrightarrow p=2,y=1$ or, $y=0,p=3,x=1$.Moon wrote:Find all prime numbers $p$ and integers $a$ and $b$ (not necessarily positive) such that $p^a + p^b$
is the square of a rational number.
Last edited by Moon on Thu Dec 09, 2010 1:57 am, edited 2 times in total.
Reason: Mod EDIT:Use \Longrightarrow instead of \implies in the LaTeX code. Try not to use text in LaTeX code, as currently LaTeX does not support line breaking.
Reason: Mod EDIT:Use \Longrightarrow instead of \implies in the LaTeX code. Try not to use text in LaTeX code, as currently LaTeX does not support line breaking.
One one thing is neutral in the universe, that is $0$.
Re: BdMO 2010 H. Sec. problem 8
Thanks for editing.Actually I posted it from mobile,so this problem occured.Hope I will be careful.
One one thing is neutral in the universe, that is $0$.
Re: BdMO 2010 H. Sec. problem 8
No problem! This forum is new. So we all will make some mistakes. BTW I won't ban you for this! 
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Re: BdMO 2010 H. Sec. problem 8
Well done!
"Everything should be made as simple as possible, but not simpler." - Albert Einstein
Re: BdMO 2010 H. Sec. problem 8
How nice solution! I am not well know to use LaTeX or any code 
so, please feel me. Dont ban me
so, please feel me. Dont ban me
A man is not finished when he's defeated, he's finished when he quits.
Re: BdMO 2010 H. Sec. problem 8
No one going to be banned unless they violate all the rules of our forum. I was just cutting a joke when in my last post.
BTW here you can find how to write LaTeX without learning LaTeX code.
Anyway, welcome to our community. Just keep posting.
BTW here you can find how to write LaTeX without learning LaTeX code.
Anyway, welcome to our community. Just keep posting.
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Re: BdMO 2010 H. Sec. problem 8
GCD means Greatest Common Divisor (গ.সা.গু)
$a\equiv b(mod\ n)$ means if you divide $a$ and $b$ by $n$, you will get same residue. In other words, $a-b$ is divisible by $n$.
You can read more here.
$a\equiv b(mod\ n)$ means if you divide $a$ and $b$ by $n$, you will get same residue. In other words, $a-b$ is divisible by $n$.
You can read more here.
Every logical solution to a problem has its own beauty.
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(Important: Please make sure that you have read about the Rules, Posting Permissions and Forum Language)