A350165 Fundamental discriminants of real quadratic number fields with odd class number > 1 whose fundamental unit has norm -1.

229, 257, 401, 577, 733, 761, 1009, 1093, 1129, 1229, 1297, 1373, 1429, 1489, 1601, 1901, 2029, 2081, 2089, 2153, 2213, 2557, 2677, 2713, 2777, 2857, 2917, 3121, 3137, 3181, 3221, 3229, 3253, 3877, 3889, 4001, 4229, 4357, 4409, 4441, 4481, 4493, 4597, 4649, 4729, 4889, 4933

Offset: 1

Author

Jianing Song, Dec 29 2021

Keywords

• nonn

Comments

Prime terms of A342368.

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. This sequence gives values for d in the case (i) and that the real quadratic number field with discriminant d has odd class number > 1.

Examples

			229 is a term since the quadratic field with discriminant 229 (Q(sqrt(229))) has class number 5. The fundamental unit of that field ((15+sqrt(229))/2) has norm -1.
401 is a term since the quadratic field with discriminant 401 (Q(sqrt(401))) has class number 5. The fundamental unit of that field (20+sqrt(401)) has norm -1.
		

Links

• Winston de Greef, 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.
• 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

Intersection of A342368 and A003653. Equals A342368 \ A349419.

Cf. A003658, A003656.

Programs

PARI

isA350165(D) = if(isprime(D) && isfundamental(D), my(h=quadclassunit(D)[1]); (h%2)&&(h>1), 0)