Find all integers p and q that satisfy p³+27pq+2009=q³.

using the following mathematica code I've varified that
|p|>1000 and |q|>1000

Fnc[x_,y_]:=x^3+27*x*y+2009-y^3;
Lmt:=1000;
Sols:={};
For[a=0,a<=Lmt,a++,
    For[b=0,b<=Lmt,b++,
        If[Fnc[a,b]==0,
          Sols=Append[Sols,{a,b}];
          ];
        If[Fnc[a,-b]==0,
          Sols=Append[Sols,{a,-b}];
          ];
        If[Fnc[-a,b]==0,
          Sols=Append[Sols,{-a,b}];
          ];
        If[Fnc[-a,-b]==0,
          Sols=Append[Sols,{-a,-b}];
          ];
        ];
    ];
Sols

also if you consider solutions of the form |x|=|y| we can show that (-7,7) is the only one

if x=y we get
27x^2+2009=0
which has no real solutions

if x=-y we get
2x^3-27x^2+2009=0
(x+7)(2x^2-41x+287)=0

first factor gives Larry's solution of (-7,7) and the second factor gives no integer solutions.


Thus (-7,7) is the only solution of the form |x|=|y|

Posted by Daniel on 2006-08-27 11:32:59