There is a puzzle today! I will solve it.
The first equation implies Y+2>=2X+1, and the second equation implies 4X>=2Y+1. Therefore, 2Y+4>=4X+2>=2Y+3. Since 4X+2 is even, 2Y+4=4X+2. That means that Y=2X1. However, Y^2+4X=Y^2+2Y+2, which is never a square for positive Y. Therefore, it is impossible to have X^2+Y+2 and Y^2+4X both be squares.

Posted by Math Man
on 20141129 10:16:30 