Four, Can't You Give Me a Better Number?

A perfect square trinomial is one that has a squared first term, a squared last term, and the middle term is twice the product of first term and last term: a^2 + 2ab + b^2.=(a+b)^2 So, the problem is to find a and b so as to maximize x. a=2^x, b=2^27 (or a=2^27 & b=2^x) This gives x=19972

Let us rewrite this expression as:

$4^{10000} + 4^{27} + 4^{x} = 2^{20000} + (2^{27})^{2} + (2^x)^{2} = (2^x + 2^{27})^2$

which requires the term $2^{20000} = 2 \cdot 2^{19999} = 2 \cdot (2^x)(2^{27}) \Rightarrow 19999 = x+27 \Rightarrow \boxed{x = 19972}.$