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. Also, I believe Gauss already knew that these 9 rings had unique factorization, and conjectured that they were the only ones; Heegner and Stark each proved this conjecture. --Dan << 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.
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