A342368 -id:A342368 - cp's OEIS frontend

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

Jianing Song, Dec 29 2021

nonn

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.

			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.

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

Intersection of A342368 and A003653. Equals A342368 \ A349419.

Cf. A003658, A003656.

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

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

Original entry on oeis.org

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

Jianing Song, Dec 29 2021

nonn

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.

			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.

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

Intersection of A342368 and A349649. Equals A342368 \ A350165.

Cf. A003658, A003656, A327297.

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

cp's OEIS Frontend

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

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