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A342368 Fundamental discriminants of real quadratic number fields with odd class number > 1.

%I A342368 #16 Feb 16 2025 08:34:01
%S A342368 229,257,316,321,401,469,473,568,577,733,761,817,892,993,1009,1016,
%T A342368 1093,1101,1129,1229,1257,1297,1304,1373,1393,1429,1436,1489,1509,
%U A342368 1601,1641,1756,1761,1772,1897,1901,1929,1957,1996,2021,2029,2081,2089,2101,2153,2177,2213
%N A342368 Fundamental discriminants of real quadratic number fields with odd class number > 1.
%C 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. A003656 gives the case where the class number is 1.
%H A342368 Jianing Song, <a href="/A342368/b342368.txt">Table of n, a(n) for n = 1..22463</a> (all terms <= 10^6).
%H A342368 Ezra Brown, <a href="https://doi.org/10.1090/S0002-9947-1974-0364172-9">Class numbers of real quadratic number fields</a>, Trans. Amer. Math. Soc. 190 (1974), 99-107.
%H A342368 Henri Cohen and X.-F. Roblot, <a href="http://dx.doi.org/10.1090/S0025-5718-99-01111-4">Computing the Hilbert Class Field of Real Quadratic Fields</a>, Math. Comp. 69 (2000), 1229-1244.
%H A342368 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ClassNumber.html">Class Number</a>
%H A342368 <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>
%e A342368 The class number of the quadratic field with discriminant 229 (namely Q(sqrt(229))) is 3, so 229 is a term.
%e A342368 The class number of the quadratic field with discriminant 1756 (namely Q(sqrt(439))) is 5, so 1756 is a term.
%o A342368 (PARI) isA342368(D) = if((D>1) && isfundamental(D), my(h=quadclassunit(D)[1]); (h%2)&&(h>1), 0)
%Y A342368 Cf. A003656.
%K A342368 nonn
%O A342368 1,1
%A A342368 _Jianing Song_, Mar 09 2021