cp's OEIS Frontend

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

%I A350165 #32 Feb 16 2025 08:34:02
%S A350165 229,257,401,577,733,761,1009,1093,1129,1229,1297,1373,1429,1489,1601,
%T A350165 1901,2029,2081,2089,2153,2213,2557,2677,2713,2777,2857,2917,3121,
%U A350165 3137,3181,3221,3229,3253,3877,3889,4001,4229,4357,4409,4441,4481,4493,4597,4649,4729,4889,4933
%N A350165 Fundamental discriminants of real quadratic number fields with odd class number > 1 whose fundamental unit has norm -1.
%C A350165 Prime terms of A342368.
%C A350165 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.
%H A350165 Winston de Greef, <a href="/A350165/b350165.txt">Table of n, a(n) for n = 1..10000</a>
%H A350165 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 A350165 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 A350165 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ClassNumber.html">Class Number</a>
%H A350165 <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>
%e A350165 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.
%e A350165 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.
%o A350165 (PARI) isA350165(D) = if(isprime(D) && isfundamental(D), my(h=quadclassunit(D)[1]); (h%2)&&(h>1), 0)
%Y A350165 Intersection of A342368 and A003653. Equals A342368 \ A349419.
%Y A350165 Cf. A003658, A003656.
%K A350165 nonn
%O A350165 1,1
%A A350165 _Jianing Song_, Dec 29 2021