A341649 Integers k such that Z[sqrt(k)] = Z[x]/(x^2 - k) is a unique factorization domain.

-2, -1, 2, 3, 6, 7, 11, 14, 19, 22, 23, 31, 38, 43, 46, 47, 59, 62, 67, 71, 83, 86, 94, 103, 107, 118, 127, 131, 134, 139, 151, 158, 163, 166, 167, 179, 191, 199, 206, 211, 214, 227, 239, 251, 262, 263, 271, 278, 283, 302, 307, 311, 331, 334, 347, 358, 367, 379

Offset: 1

Author

Jianing Song, Feb 16 2021

Keywords

• sign

Comments

Equivalently, integers k such that Z[sqrt(k)] = Z[x]/(x^2 - k) is a principal ideal domain.

-2, -1, together with k such that 4*k is in A003656.

All terms are squarefree and congruent to 2 or 3 modulo 4. It appears that the terms > 2 are of the form p or 2*p, where p is a prime congruent to 3 modulo 4. [This is correct; see Theorem 1 and Theorem 2 of Ezra Brown's link. - Jianing Song, Feb 24 2021]

The smallest prime p == 3 (mod 4) that is not a term is p = 79. The smallest prime p == 3 (mod 4) such that 2*p is not a term is p = 71.

Examples

			Z[sqrt(-1)] = Z[i] is the ring of Gaussian integers, which is a unique factorization domain.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.
• Mathematics Stack Exchange, Unique factorization domain that is not a Principal ideal domain
• Eric Weisstein's World of Mathematics, Class Number
• Index entries for sequences related to quadratic fields

Crossrefs

Cf. A003656, A002052 (odd primes in this sequence).

Programs

PARI

isA341649(n) = my(D=4*n); isfundamental(D) && quadclassunit(D)[1] == 1