J.L.Masley & H.M.Montgomery 1975: if the Nth roots of unity (i.e. complex R such that R^N=1) are adjoined to the integers, then we get a ring enjoying "unique factorization into primes" exactly in the following cases: N = 1,3,4,5,7,8,9,11,12,15,16,20,24, 13,17,19,21,25,27,28,32,33,35,36,40,44,45,48,60,84. K.Heegner 1952 & H.Stark 1967: If sqrt(-N) is adjoined to the integers, N>0, then we get a ring enjoying "unique factorization into primes" exactly in the following cases: N = 1,2,3,7,11, 19,43,67,163. In both cases, for the N in the first line there is a known Euclid-like GCD algorithm which proves it. For the N in the second line, the proof is harder. The proof that no other N work is way harder. For example 6 = 2*3 = (1+i*sqrt5)*(1-i*sqrt5) has nonunique factorization for N=5. Exactly for N = 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 and -1,-2,-3,-7,-11, adjoining sqrt(N) to the integers yields a norm-lowering Euclidean algorithm. V.Cioffari 1979: Exactly for N = 2,3,10, adjoining cbrt(N) to the integers yields a norm-lowering Euclidean algorithm. [Also known we get finite sets for 4th roots and 5th roots.] Exactly for (A,B) in the following list, adjoining both sqrt(A) and sqrt(B) to the integers yields a ring with a norm-lowering Euclidean algorithm: (-1, 2), (-1, 3), (-1, 5), (-1, 7), (-2, -3), (-2, 5), (-3, 2), (-3, 5), (-3,-7), (-3, -11), (-3, 17), (-3, -19), (-7, 5) [F.Lemmermeyer 1989-1995] There are only finitely many cubic totally-complex or totally-real-cyclic algebraic number fields with a norm-lowering Euclidean algorithm (H.Davenport 1950; H.Heilbronn 1938). "Totally complex" means none of the roots of the minimal polynomial are real. Totally real means all of them are real. Here "cyclic" means that if you adjoin one root of the minimal polynomial, that is equivalent to adjoining all of them. Almost all cubic real fields are noncyclic. There are only finitely many quartic totally-complex and totally-real algebraic number fields with a norm-lowering Euclidean algorithm (H.Davenport 1950; F.Lemmermeyer 1995, D.A.Clark 1992-6). In all of the above cases, the list of rings was finite. Can we make some such list of N which is INFINITE? Apparently the answer is yes, but I'm not sure anybody has ever proved it in any interesting sense? Adjoining pi to the integers has unique factorization but it is boring/useless because it is nondiscrete and there is no way to multiply to ring elements to get an integer (aside from the obvious). Here are two apparently-infinite lists of N such that adjoining sqrt(N) to the integers yields a ring enjoying unique factorization http://oeis.org/A003172 http://oeis.org/A003655 but in these cases (a) I don't know if anybody ever proved they really are infinite -- I think the question is open -- albeit see H. te Riele & H.Williams: http://www.emis.de/journals/EM/expmath/volumes/12/12.1/pp99_113.pdf for large computations suggesting infinity; (b) even if they are infinite, I think it isn't very interesting/useful unless you have a Euclid algorithm so you can rapidly find GCD, and a notion of norm so we can talk about A being "smaller than" B and thus the "least" factor. These lists if restricted to such cases become finite. So... QUESTION: So what I want, but do not have, is an infinite set of ways to extend the integers into a larger countable ring (but still contained inside the reals or complex numbers or some finite-dimensional -- preferably bounded dimensional -- such space) so that (i) we get unique factorization via norm-based Euclidean algorithm. (ii) we get discreteness, i.e. the number of ring elements in any ball in the complex plane, or anyhow within some finite-dimensional space erected over the integers, is finite. This seems to force us to restrict to ALGEBRAIC number fields. Heilbronn 1938 and/or 1950 conjectured the number of cubic non-cyclic totally-real algebraic number fields with a norm-lowering Euclidean algorithm, was infinite. This conjecture if true would seem to solve my problem. However, I think this conjecture still is open, albeit seems supported by computer evidence, Lemmermeyer: http://www.rzuser.uni-heidelberg.de/~hb3/publ/survey.pdf