Re: Infinite solutions proof

A much easier solution:

For example, (x,y,z) = (3,4,5) works. We can multiply x,y,z by the same integer constant yielding (x,y,z) = (6,8,10), (9,12,15), etc. Hence there are infinitely many solutions.

Re: Infinite solutions proof

I'm supposed to use this method. The only part I'm unsure of is where I derive 2=2. Did I do this properly?

Re: Infinite solutions proof

Ah okay. Your solution is correct, but too long. You can just stop when you know both sides of the equation are equal, i.e. $\displaystyle m^4 + n^4 + 2(mn)^2 = m^4 + n^4 + 2(mn)^2$. This equality implies that the above holds regardless of your choice of m and n, as long as m > n (so that x is positive).

Re: Infinite solutions proof

So, I guess I could say that there are infinite solutions because both sides of the equation are identical. That is, as long as m > n.