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

1,1

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.

Links

Table of n, a(n) for n=1..46.

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

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.

Prog

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

Crossrefs

Intersection of A342368 and A349649. Equals A342368 \ A350165.

Cf. A003658, A003656, A327297.

Sequence in context: A231760 A048905 A200314 * A115559 A033861 A179155

Adjacent sequences: A349416 A349417 A349418 * A349420 A349421 A349422

Keyword

nonn,changed

Author

Jianing Song, Dec 29 2021

Status

approved