Number theory begins by investigating properties of the integers. Analogous properties can also be asked about polynomials. For instance, we can consider Fermat's last theorem in its classical form (are there positive integer solutions to xⁿ + yⁿ = zⁿ when n > 2?) and for polynomials (can we solve f(t)ⁿ + g(t)ⁿ = h(t)ⁿ in polynomials?) We will also discuss the abc conjecture for the integers and for polynomials, give a proof of the conjecture in the polynomial case, and use that to give a proof of Fermat's last theorem for polynomials. We will then explore other uses for the abc conjecture.

Finally, we will consider a generalization of the abc conjecture and, if time permits, give the outline of a proof in the polynomial case.