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A341649 Integers k such that Z[sqrt(k)] = Z[x]/(x^2 - k) is a unique factorization domain.

%I A341649 #20 Feb 16 2025 08:34:01
%S A341649 -2,-1,2,3,6,7,11,14,19,22,23,31,38,43,46,47,59,62,67,71,83,86,94,103,
%T A341649 107,118,127,131,134,139,151,158,163,166,167,179,191,199,206,211,214,
%U A341649 227,239,251,262,263,271,278,283,302,307,311,331,334,347,358,367,379
%N A341649 Integers k such that Z[sqrt(k)] = Z[x]/(x^2 - k) is a unique factorization domain.
%C A341649 Equivalently, integers k such that Z[sqrt(k)] = Z[x]/(x^2 - k) is a principal ideal domain.
%C A341649 -2, -1, together with k such that 4*k is in A003656.
%C A341649 All terms are squarefree and congruent to 2 or 3 modulo 4. It appears that the terms > 2 are of the form p or 2*p, where p is a prime congruent to 3 modulo 4. [This is correct; see Theorem 1 and Theorem 2 of Ezra Brown's link. - _Jianing Song_, Feb 24 2021]
%C A341649 The smallest prime p == 3 (mod 4) that is not a term is p = 79. The smallest prime p == 3 (mod 4) such that 2*p is not a term is p = 71.
%H A341649 Jianing Song, <a href="/A341649/b341649.txt">Table of n, a(n) for n = 1..10000</a>
%H A341649 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 A341649 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/16754/unique-factorization-domain-that-is-not-a-principal-ideal-domain">Unique factorization domain that is not a Principal ideal domain</a>
%H A341649 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ClassNumber.html">Class Number</a>
%H A341649 <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>
%e A341649 Z[sqrt(-1)] = Z[i] is the ring of Gaussian integers, which is a unique factorization domain.
%o A341649 (PARI) isA341649(n) = my(D=4*n); isfundamental(D) && quadclassunit(D)[1] == 1
%Y A341649 Cf. A003656, A002052 (odd primes in this sequence).
%K A341649 sign
%O A341649 1,1
%A A341649 _Jianing Song_, Feb 16 2021