A342368 Fundamental discriminants of real quadratic number fields with odd class number > 1.

229, 257, 316, 321, 401, 469, 473, 568, 577, 733, 761, 817, 892, 993, 1009, 1016, 1093, 1101, 1129, 1229, 1257, 1297, 1304, 1373, 1393, 1429, 1436, 1489, 1509, 1601, 1641, 1756, 1761, 1772, 1897, 1901, 1929, 1957, 1996, 2021, 2029, 2081, 2089, 2101, 2153, 2177, 2213

Offset

1,1

Comments

For a positive fundamental discriminant d, the class number of the real quadratic field of discriminant d is odd if and only if d = 8 or is of one of the three following forms: (i) p, where p is a prime congruent to 1 modulo 4; (ii) 4p or 8p, where p is a prime congruent to 3 modulo 4; (iii) pq, where p, q are distinct primes congruent to 3 modulo 4. See Theorem 1 and Theorem 2 of Ezra Brown's link. A003656 gives the case where the class number is 1.

Links

Jianing Song, Table of n, a(n) for n = 1..22463 (all terms <= 10^6).

Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.

Henri Cohen and X.-F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244.

Eric Weisstein's World of Mathematics, Class Number

Index entries for sequences related to quadratic fields

Example

The class number of the quadratic field with discriminant 229 (namely Q(sqrt(229))) is 3, so 229 is a term.
The class number of the quadratic field with discriminant 1756 (namely Q(sqrt(439))) is 5, so 1756 is a term.

Prog

(PARI) isA342368(D) = if((D>1) && isfundamental(D), my(h=quadclassunit(D)[1]); (h%2)&&(h>1), 0)

Crossrefs

Cf. A003656.

Sequence in context: A140017 A119711 A062589 * A250235 A250236 A094612

Adjacent sequences: A342365 A342366 A342367 * A342369 A342370 A342371

Keyword

nonn

Author

Jianing Song, Mar 09 2021

Status

approved