A349419 Fundamental discriminants of real quadratic number fields with odd class number > 1 whose fundamental unit has norm 1.

316, 321, 469, 473, 568, 817, 892, 993, 1016, 1101, 1257, 1304, 1393, 1436, 1509, 1641, 1756, 1761, 1772, 1897, 1929, 1957, 1996, 2021, 2101, 2177, 2429, 2589, 2636, 2908, 2913, 2981, 3173, 3261, 3356, 3569, 3736, 3873, 3928, 3941, 3957, 3981, 3997, 4009, 4193, 4281

Offset: 1

Author

Jianing Song, Dec 29 2021

Keywords

• nonn

Comments

Composite 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 cases (ii) and (iii) and that the real quadratic number field with discriminant d has odd class number > 1.

Examples

			316 is a term since the quadratic field with discriminant 316 (Q(sqrt(79))) has class number 3. The fundamental unit of that field (80+9*sqrt(79)) has norm 1.
321 is a term since the quadratic field with discriminant 321 (Q(sqrt(321))) has class number 3. The fundamental unit of that field (215+12*sqrt(321)) has norm 1.
		

Links

• 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 A349649. Equals A342368 \ A350165.

Cf. A003658, A003656, A327297.

Programs

PARI

isA349419(D) = if(!isprime(D) && (D>1) && isfundamental(D), my(h=quadclassunit(D)[1]); (h%2)&&(h>1), 0)