In Chaitin's Meta Math! The Quest For Omega, he briefly talks about Hilbert's 10th Problem. He then says that any Diophantine Equation $p=0$ can be changed into two equal polynomials with positive integer coefficients: $p=0 \iff p_1 = p_2$.
Then he says that we can think of these equations like a "computer":
Diophantine Equation Computer: $$L(k,n,x,y,z,...)=R(k,n,x,y,z,...)$$ Program: $k$, Output: $n$, Time: $x,y,z,...$
With left side $L$, right side $R$. He says $k$ is the program of this computer, which outputs $n$. He also says that the unknowns are a multi-dimensional time variable.
What confuses me is that he then says that Hilbert's 10th Problem is clearly not solvable when viewed in this way. He basically says "because of Turing's Halting Problem". But I don't see the connection (I'm just beginning to learn the theory). I was hoping someone could more clearly explain what Chaitin's point is here.
I know that Turing's Halting Problem basically states that you cannot predict when a program will halt before it actually halts (given a finite amount of time). What is the application to Hilbert's 10th Problem, using the notation laid out by Chaitin?