New Proofs Probe the Limits of Mathematical Truth

Something that may not be clear when reading this is the distinction between complex numbers and the ring of integers adjoined with i.

"Complex numbers" are of the form a+bi where a and b can be any real number -- 1, 5, pi, the square root of 2, -2.14, etc.

The ring of integers adjoined with i are numbers of the form a+bi where a and b are both integers (-1, 5, 34, etc).

You can also, in addition to using i, adjoin any real number to the integers and get a new field with numbers of the form a+bx where a and b are integers and x is any additional number you want to add-- frequently square roots like the square root of two.

This result shows undecidability of diophantine equations in all those fields of integers, but not complex numbers, for which it's easy to prove that there are _always_ solutions.