A327297 Terms in A003656 that are not prime powers (A246655).

12, 21, 24, 28, 33, 44, 56, 57, 69, 76, 77, 88, 92, 93, 124, 129, 133, 141, 152, 161, 172, 177, 184, 188, 201, 209, 213, 217, 236, 237, 248, 249, 253, 268, 284, 301, 309, 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

Offset: 1

Author

Jianing Song, Sep 16 2019

Keywords

• nonn

Comments

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]

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:

(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)).

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

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

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]

Links

• Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.

Crossrefs

Subsequence of A003656 and A003658.

Complement of A003655 with respect to A003656.

Cf. A016105, A246655.

Programs

PARI

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