A355461 Squarefree numbers d of the form r^2*m^2 + 4*r, where r and m are odd positive integers, such that Q(sqrt(d)) has class number 1.

5, 13, 21, 29, 53, 173, 237, 293, 437, 453, 1133, 1253

Offset: 1

Author

Marco Ripà, Jul 02 2022

Keywords

• nonn
• fini
• full

Comments

In 1801, Gauss conjectured that there exist infinitely many real quadratic fields with class number one and the conjecture is still unproved, but there are only 12 real quadratic fields of class number one which are of the form Q(sqrt(r^2*m^2 + 4*r)), where the parameters r and m are odd integers. Those 12 values of d := r^2*m^2 + 4*r belong to the present sequence.

Examples

			a(2) = 13 since h(13) = h(1^2*3^2 + 4*1) = 1.
		

Links

• A. Biró and K. Lapkova The class number one problem for the real quadratic fields Q(sqrt(a*n^2+4*a)), Acta Arith., vol. 172(2), 2016, pp. 117-131.
• A. Hoque and S. Kotyada Class number one problem for the real quadratic fields Q(sqrt(m^2+2*r)), Archiv der Mathematik, vol. 116(1), 2021, pp. 33-36.

Crossrefs

Cf. A050950, A053329, A308420.