I am not even real sure what the second problem is actually asking me to do. Any help would be appreciated. All eight problems of this last chapter in my Intro to Number Theory class seem incredibly hard to me. I will post a reply with my hopeful solution to the first problem which I think I have at least partially figured out.

#1 (solved)

Show that the equation

$\displaystyle x^4 + 4y^4 = z^2,\ \ \ \ \ x \neq 0, y \neq 0, z \neq 0$

has no solutions. It may be helpful to reduce this to the case that x > 0, y > 0, z > 0, (x,y)=1, and then by dividing by 4 (if necessary) to further reduce this to where x is odd.

#2

Show that there is no right triangle with integral sides whose area is a perfect square by showing that it suffices to work with primitive triangles and with them, one is led to the equation given in problem 1.

and #3 (solved)...I printed out the latex help sheet, but i do not see a 'not equal' sign anywhere on it. Or a not congruent sign for that matter. I typed it out for now, but can anyone tell me what is proper to use for that? I will search the latex forum for it later.