3. Alice is playing a FactorFinding game with herself. The integer factorization is not exciting enough so she decides to consider another number system. - Alice thinks about the "complex integers", i.e. a +bi where a, b are integers and i example, addition: (1 + 2i) + (3 + 4i) = 4 + 6i; multiplication: (1 + 2i)(3+4i) = 38+ (6 + 4)i = 5+ 10i -1. For She then plays the game as follows. Alice starts with an arbitrary "complex integer a +bi and k 1. Alice tries to find ak+1+bk+1i that divides ak +bki which means there exist two integers c and d such that (ak+1 + bk+1i)(c + di) = ak + bki. If |c| + |d| 1, then the game terminates. Otherwise, she increments k and repeats this step. Question: Can Alice continue the game forever? If yes, give an example infinite run. If not, prove it by the potential-function method. Hint: Try the potential function (a + bi) = a + 6. Solution: