A308420 Squarefree numbers d of the form s^2 + r, where r divides 4s, such that Q(sqrt(d)) has class number 1.

2, 3, 5, 6, 7, 11, 13, 14, 17, 21, 23, 29, 33, 37, 38, 47, 53, 62, 69, 77, 83, 93, 101, 141, 167, 173, 197, 213, 227, 237, 293, 398, 413, 437, 453, 573, 677, 717, 1077, 1133, 1253, 1293, 1757

Offset

1,1

Comments

This sequence is finite, but might not be given in full if the generalized Riemann hypothesis is false.

References

Richard A. Mollin, Quadratics. p. 176, Theorem 5.4.3. Given "a fundamental discriminant of ERD-type with radicand D," the ring of Q(sqrt(D)) has class number 1 "if and only if D" is one of the values listed above, "with one possible exceptional value whose existence would be a counterexample to the GRH" (generalized Riemann hypothesis).

Links

Table of n, a(n) for n=1..43.

Formula

Given d = s^2 + r where r | 4s (this is called "extended Richaud-Degert type" or "ERD-type" by Mollin), d is in this sequence if h(O_Q(sqrt(d))) = 1, where h(O_K) is the class number of the ring of algebraic integers O_K.

Example

Since 7 = 3^2 - 2 (note that 2 is a divisor of 4 * 9) and h(Z[sqrt(7)]) = 1, 7 is in the sequence.
Although 10 = 3^2 + 1, we see that h(Z[sqrt(10)]) > 1 since Z[sqrt(10)] is not a unique factorization domain (e.g., 10 = 2 * 5 = sqrt(10)^2). So 10 is not in the sequence.
Although h(Z[sqrt(19)]) = 1, there is no way to express 19 as s^2 + r, e.g., 19 = 3^2 + 10 but 10 is not a divisor of 12, 19 = 4^2 + 3 but 3 is not a divisor of 16, 19 = 5^2 - 6 but 6 is not a divisor of 20, 19 = 6^2 - 17 but -17 does not divide 24. So 19 is not in the sequence either.

Crossrefs

Cf. A053329 (first differs at the 16th term), A003172.

Sequence in context: A380525 A380468 A053329 * A363462 A379948 A260442

Adjacent sequences: A308417 A308418 A308419 * A308421 A308422 A308423

Keyword

nonn,fini

Author

Alonso del Arte, May 26 2019

Status

approved