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
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
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
KEYWORD
nonn
AUTHOR
Jianing Song, Dec 29 2021
STATUS
approved