Let be a Euclidean Domain.
- Prove that if and divides , then divides . More generally, show that if divides with , then divides .
- Consider the Diophantine Equation , where , , and are integers and . Suppose is a solution- that is, . Prove that the full set of solutions to this equation is given by and as ranges over the integers.
- Suppose first that and that . By Theorem 4, there exist such that . Then . Since , we have for some . So , and thus . So .
More generally, say , , and . Again by Theorem 4, we have such that . Since , we have for some . Now , so that , and thus Since is an integral domain, , and thus divides .
- Write , and suppose is a solution to . Then , so that . Since divides , we have that divides . Say ; then . Substituting into , we see that , so that . Since divides and is a domain, we have .
Moreover, it is straightforward to show (as we did in this previous exercise) that every pair of this form is in fact a solution. Thus we have completely characterized the solutions of .