BdMO 2010 H. Sec. problem 8

[rakeen]

And what would be "law sha gu"
What a "residue"? And thnx for the link

[rakeen]

thnx a lot. And wot does "law sha gu" defers? Whatz a "residue"?
Thnx for the link. Another 1. What is modular arithmatic?(silly isn't it)

[HandaramTheGreat]

ল.সা.গু.=Least Common Multiple=LCM
residue=ভাগশেষ

[rakeen]

thnx. But what abt "law sha gu" ? and whatz a 'residue'? o! another 1 : modular arithmatic ki?(silly isn't it)

[Masum]

See here:
http://en.wikipedia.org/wiki/Congruence
http://mathworld.wolfram.com/Congruence.html
http://mathworld.wolfram.com/topics/Congruences.html
And a general suggestion from me,when you get fixed with such problems,search in google.This helps too much

[Hasib]

From one defination of mod ($\equiv$) it's very easy to find the other defination. Try yourself!

[rakeen]

I really understood these stuff now. thnx everyboody. But still cant solve some problems relating to MA

[Masum]

Then why don't you post them?

[sm.joty]

মাসুম ভাইয়ের সমাধান বুঝি নাই। কেউ একটু ব্যাখ্যা করে বোঝান, উনি তো অনেক শর্ট কার্ট করছেন।

[Masum]

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.

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, $y=0,p=3,x=1$.

You can ask me for an exact part's explanation or the whole solution with details