cp's OEIS Frontend

A236307 Discriminants d such that the ring of algebraic integers of Q(sqrt(-d)) is not a unique factorization domain.

%I A236307 #27 Dec 27 2018 22:05:35
%S A236307 5,6,10,13,14,15,17,21,22,23,26,29,30,31,33,34,35,37,38,39,41,42,46,
%T A236307 47,51,53,55,57,58,59,61,62,65,66,69,70,71,73,74,77,78,79,82,83,85,86,
%U A236307 87,89,91,93,94,95,97,101,102,103,105,106,107,109,110,111,113,114,115,118,119,122
%N A236307 Discriminants d such that the ring of algebraic integers of Q(sqrt(-d)) is not a unique factorization domain.
%C A236307 Stewart & Tall (2002) show that none of the first thirteen terms listed here correspond to an imaginary quadratic ring with unique factorization by giving one example of an integer having two distinct factorizations for each ring.
%C A236307 This sequence consists of the squarefree numbers (A005117) that are not Heegner numbers (A003173).
%D A236307 Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts: A. K. Peters (2002): p. 83, Theorem 4.10.
%H A236307 Steven R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a>, p. 5, Table 2. [Cached copy, with permission of the author]
%F A236307 a(n) = A005117(n + 9) for n > 91.
%e A236307 10 is in the sequence because 14 = 2 * 7 = (2 - sqrt(-10))(2 + sqrt(-10)), which are two distinct factorizations of 14 in Z[sqrt(-10)].
%e A236307 13 is in the sequence because 14 = 2 * 7 = (1 - sqrt(-13))(1 + sqrt(-13)), which are two distinct factorizations of 14 in Z[sqrt(-13)].
%e A236307 14 is in the sequence because 15 = 3 * 5 = (1 - sqrt(-14))(1 + sqrt(-14)), which are two distinct factorizations of 15 in Z[sqrt(-14)].
%e A236307 (Many more examples can be found for each ring; these three are from the thirteen given by Stewart & Tall (2002)).
%e A236307 And when -d = 1 mod 4 other than -3, -7, -11, -19, -43, -67 or -163, we can often use (d + 1)/4 = (1/2 - sqrt(-d)/2)(1/2 + sqrt(-d)/2) as an example, such as 4 = 2 * 2 = (1/2 - sqrt(-15)/2)(1/2 + sqrt(-15)/2) in O_(Q(sqrt(-15))).
%t A236307 Select[Range[100], SquareFreeQ[#] && NumberFieldClassNumber[Sqrt[-#]] > 1 &]
%Y A236307 Cf. A005117, A003173, A005847, A006203, A046085.
%K A236307 nonn
%O A236307 1,1
%A A236307 _Alonso del Arte_, Apr 21 2014
%E A236307 Name corrected after an e-mail from _Michel Lagneau_, Dec 25 2018