Last time, we talked about the failure of prime factorization. The set of all numbers of the form (a + b*sqrt(-5)), for example, form a ring, Z[sqrt(-5)]. In this ring, the number 6 factors into 2 and 3 as well as (1-sqrt(-5)) and (1+sqrt(-5)). So unique prime factorization fails because these last two numbers cannot be reduced further.
Now a question is, what negative numbers put inside the square results in unique prime factorization failure? Kurt Heegner was the one who first proposed that only 9 numbers result in NO failure. 1,2,3,7,11,19,43,67,163. No failure means that we still have a unique factorization domain. The expanded ring Z[sqrt(-1)] or Z[sqrt(-2)] still has unique prime factorization for all elements in the ring.
