A308420 -id:A308420 - cp's OEIS frontend

A355424 Positive integers m such that the real quadratic fields of the form Q(sqrt(m^2+4)) have class number 1.

1, 3, 5, 7, 13, 17

Offset: 1

Marco Ripà, Jul 01 2022

Former Yokoi's conjecture, proved by Biró in 2003 (see References). There are only six real quadratic fields of the form Q(sqrt(a(n)^2+4)), where Q indicates the set of rational numbers, with class number one.

			a(1) = 1, since h(1^2 + 4) = h(5) = 1.

A. Biró, Yokoi's conjecture, Acta Arith., vol. 106(1), 2003, pp. 85-104.

Cf. A050950, A053329, A308420.

Let n be a positive integer less than 7. a(n) = 4*n - 7 iff n = 5, 6 and a(n) = 1 + 2*(n - 1) otherwise.

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

Original entry on oeis.org

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

Offset: 1

Marco Ripà, Jul 02 2022

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.

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

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.

Cf. A050950, A053329, A308420.

cp's OEIS Frontend

A355424 Positive integers m such that the real quadratic fields of the form Q(sqrt(m^2+4)) have class number 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

cp's OEIS Frontend

A355424 Positive integers m such that the real quadratic fields of the form Q(sqrt(m^2+4)) have class number 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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