16.1

1. Why do we not have to consider a quadratic term in the x's? Is it possible to write any elliptic curve without that term by using a method similar to completing the square? Also, isn't it possible to choose two points so that a third point isn't intersected? What then?

2. I have never been exposed to elliptic curves before, at least not in an expository setting. Thinking about this for a moment, I bet that any elliptic curve with rational coefficients can be obtained through a rational map with integer coefficients. I have discovered a truly remarkable proof for this, but it is too large to put in this blog post. Seriously though, it is pretty incredible that such seemingly unrelated fields (algebra and analysis? it's all numbers anyway) could be linked in such an elegant way. And to think that that link somehow proves that x^n+y^n=z^n has no integer solutions for n>2. It's amazing.