It's a string of 0s and 1s in which each digit is as unrelated to its predecessor as one coin toss is from the next. The same process that led Turing to conclude that the halting problem is undecidable also led Chaitin to the discovery of an unknowable number. An unknowable number wouldn't be a problem if it never reared its head. But once Chaitin had discovered Omega, he began to wonder whether it might have implications in the real world. So he decided to search mathematics for places where Omega might crop up.
Number theory is the foundation of pure mathematics.
It describes how to deal with concepts such as counting, adding, and multiplying. Chaitin formulated a Diophantine equation that was 200 pages long and had 17,000 variables. Given an equation like this, mathematicians would normally search for its solutions. But Chaitin didn't look for specific solutions, he simply looked to see whether there was a finite or an infinite number of them. He did this because he knew it was the key to unearthing Omega.