Does there exist any positive integral solution of the equation given below?

16(X³+Y³) + 12(X²+Y²) + 3(X+Y) = 2,138,156,388

First we have to look past the red herrings. The might become evident by factoring 2,138,156,388, but I just guessed that there might be a solution if I...

Let X=-Y

Then the first and third terms disappear and

12(2X²) = 2,138,156,388 

24X²=2,138,156,388 (same as 24Y²=2,138,156,388)

This reduces to X and Y=±9439

Solutions exist at (9439,-9439) and (-9439,9439)

Edited on December 28, 2005, 7:48 pm

Posted by Eric on 2005-12-28 19:37:31