In this talk, I will present a proof (due to Yuri Matiyasevich, Julia Robinson, Martin Davis, and Hilary Putnam) that for any algorithm with integer inputs, there is a corresponding integer polynomial $D(a_1,\ldots,a_n,x_1,\ldots,x_m)$ such that the algorithm halts on input $a_1, \ldots, a_n$ iff there are integers $x_1,\ldots,x_m$ such that $D(a_1,\ldots,a_n,x_1,\ldots,x_m) = 0$. In essence, this allows us to implement the operation of a computer in the simple number-theoretic issue of whether or not a polynomial has an integer solution. The proof will come in two parts: an implementation of a Turing machine using number theoretic techniques and a discussion on why Turing machines are believed to be capable of implementing any algorithm.