From: Dan Asimov These rings are defined as the algebraic integers O_N of the field Q(sqrt(-N)).
But in fact O_3 = Z[(sqrt(-3)+1)/2] and O_7 = Z[(sqrt(-7)+1)/2].
In Z[sqrt(-3)] we have (1+sqrt(-3))(1-sqrt(-3) = 2*2, and in Z[sqrt(-7)] we have (1+sqrt(-7))(1-sqrt(-7) = 2*2*2, showing non-unique factorization.
--aha, thank you for saving me from this error, the rings are not defined the way I naively thought. There may be other cases of my naivete of this ilk in there too (?). Anyway, the BIG QUESTION I wanted to stick in your faces was, can we get an INFINITE SET of these normed-Euclidean rings? I mean, I don't see why this question has to be so hard it's been open for 80-200 years. What's the big deal? If there's a big infinite pile of these fields like the computers and Heilbronn say, then why can't you just invent some unified Euclidean algorithm for them, or anyhow enough of them, and game over?