login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Terms in A003656 that are not prime powers (A246655).
3

%I #22 Dec 29 2021 02:23:44

%S 12,21,24,28,33,44,56,57,69,76,77,88,92,93,124,129,133,141,152,161,

%T 172,177,184,188,201,209,213,217,236,237,248,249,253,268,284,301,309,

%U 329,332,341,344,376,381,393,412,413,417,428,437,453,472,489,497,501,508,517,524,536,537,553,556,573,581,589,597

%N Terms in A003656 that are not prime powers (A246655).

%C Conjecture: if D is a term of this sequence, then D = uv, where u, v are 4, 8 or primes congruent to 3 modulo 4. For example, a(1) = 12 = 3*4, a(2) = 21 = 3*7, a(3) = 24 = 3*8, a(4) = 28 = 4*7, a(5) = 33 = 3*11, ... [This conjecture is correct: see Theorem 1 and Theorem 2 of Ezra Brown link; see also A003656. - _Jianing Song_, Dec 28 2021]

%C Let k be the quadratic field with discriminant D, O_k be ring of integers of k, N(x) be the norm of x and (D/p) be the Kronecker symbol. If D is a term of this sequence and D = uv, where u, v are 4, 8 or primes congruent to 3 modulo 4, then:

%C (a) if (((-u)/p), ((-v)/p)) = (1, 1), (1, 0) or (0, 1), then N(x) = p has solutions in O_k, while N(y) = -p has no solutions in k. For example, for D = 21 and p = 37, we have ((-3)/37) = ((-7)/37) = 1, and N(x) = 37 has solution x = (13 + sqrt(21))/2, but N(y) = -37 has no solutions in Q(sqrt(21)).

%C (b) if (((-u)/p), ((-v)/p)) = (-1, -1), (-1, 0) or (0, -1), then N(x) = -p has solutions in O_k, while N(y) = p has no solutions in k. For example, for D = 12 and p = 11, we have ((-3)/11) = ((-4)/11) = -1, and N(x) = -11 has solution x = 1 + 2*sqrt(3), but N(y) = 11 has no solutions in Q(sqrt(3)).

%C (c) if (((-u)/p), ((-v)/p)) = (1, -1) or (-1, 1), then N(x) = +-p has no solutions in k.

%C The smallest number of the form above that is not in this sequence is 316 = 4*79.

%C Also, it is conjectured that the quadratic field with discriminant D has form class number 2, where D is a term of this sequence. This is equivalent to the conjecture above. [This can also be deduced from the first paragraph of Ezra Brown link: the norm of the fundamental unit of the field k is -1 if D = 8 or a prime congruent to 1 modulo 4, and 1 if D is in this sequence. Here k is the quadratic field with discriminant D. - _Jianing Song_, Dec 28 2021]

%H 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.

%o (PARI) isA327297(D) = if(D>1&&isfundamental(D), quadclassunit(D)[1]==1&&!isprimepower(D), 0)

%Y Subsequence of A003656 and A003658.

%Y Complement of A003655 with respect to A003656.

%Y Cf. A016105, A246655.

%K nonn

%O 1,1

%A _Jianing Song_, Sep 16 2019