[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include(/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php) [function.include]: failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include() [function.include]: Failed opening '/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php' for inclusion (include_path='.:/opt/php53/lib/php')
[phpBB Debug] PHP Warning: in file [ROOT]/includes/session.php on line 1042: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4786: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4788: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4789: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4790: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
BdMO Online Forum • View topic - Equation

Equation

For discussing Olympiad Level Number Theory problems

Equation

x^2+xy+y^2=(x+y+3)^3/27,find all(x,y)
aritra barua

Posts: 45
Joined: Sun Dec 11, 2016 2:01 pm

Re: Equation

aritra barua wrote:x^2+xy+y^2=(x+y+3)^3/27,find all(x,y)

It is obvious that \$x+y\$ is divisible by \$3\$ . Put \$x+y\$ = \$3k\$ .
So, the equation becomes,
\$\$(3k)^2 - x(3k-x) = (k+1)^3 \$\$
=> \$\$ x^2 - 3kx + 9k^2 = k^3 + 3k^2 + 3k +1 \$\$
=> \$\$ x^2 - 3kx - k^3 + 6k^2 - 3k -1 = 0 \$\$
As we have to find the integer roots of this equation, the discriminant must be a perfect square.
Discriminant = \$\$ 9k^2 + 4k^3 - 24k^2 + 12k +4 \$\$ = \$\$ (k-2)^2 (4k+1) \$\$
So, \$4k+1\$ must be a perfect square => \$\$k = n^2 + n\$\$
Substituting this, we get 2 pairs of \$(x,y)=(n^3+3n^2-1,-n^3+3n+1)\$ and \$(-n^3 + 3n +1,n^3 + 3n^2 -1)\$

Atonu Roy Chowdhury

Posts: 40
Joined: Fri Aug 05, 2016 7:57 pm