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

5, 6, 10, 13, 14, 15, 17, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 122

Offset: 1

Author

Alonso del Arte, Apr 21 2014

Keywords

• nonn

Comments

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.

This sequence consists of the squarefree numbers (A005117) that are not Heegner numbers (A003173).

Examples

			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)].
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)].
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)].
(Many more examples can be found for each ring; these three are from the thirteen given by Stewart & Tall (2002)).
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))).
		

References

• 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.

Links

• Steven R. Finch, Class number theory, p. 5, Table 2. [Cached copy, with permission of the author]

Crossrefs

Cf. A005117, A003173, A005847, A006203, A046085.

Programs

Mathematica

Select[Range[100], SquareFreeQ[#] && NumberFieldClassNumber[Sqrt[-#]] > 1 &]

Formula

a(n) = A005117(n + 9) for n > 91.

Extensions

Name corrected after an e-mail from Michel Lagneau, Dec 25 2018