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

Author

Jianing Song, Mar 09 2021

Keywords

• nonn

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.

Examples

			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.
		

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

Crossrefs

Cf. A003656.

Programs

PARI

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