A050950 -id:A050950 - cp's OEIS frontend

A050967 Discriminants of real quadratic fields with class number 1 and related continued fraction period length of 18.

139, 163, 283, 417, 419, 566, 633, 737, 758, 781, 787, 998, 1141, 1142, 1163, 1286, 1307, 1337, 1461, 1718, 1829, 1931, 2243, 2537, 2653, 2966, 2973, 3013, 3117, 3629, 3713, 4061, 4269, 4541, 4781, 6629, 6717, 7037, 7133, 7181, 8013, 8157, 8197, 8301, 8777, 9957, 10277, 10493, 11429, 11957, 12293, 13373, 13917, 16373, 18653, 18813, 18893, 20597, 23597, 24173, 26837, 30917

Offset: 1

N. J. A. Sloane, Jan 04 2000

R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A.

R. A. Mollin and H. C. Williams, On a determination of real quadratic fields of class number one and related continued fraction period length less than 25, Proc. Japan Acad. Ser. A Math. Sci. Volume 67, Number 1 (1991), 20-25. (cf. Table 3.1)

Cf. A050950-A050969, A051962-A051965.

a(42) and following from Georg Fischer, Sep 20 2021

A050968 Discriminants of real quadratic fields with class number 1 and related continued fraction period length of 19.

Original entry on oeis.org

241, 313, 449, 829, 953, 1069, 1193, 1213, 1697, 2381, 3853, 4733, 5077, 5189, 5381, 5669, 5981, 6173, 6277, 6389, 6397, 6917, 7717, 7757, 7877, 8237, 9973, 10037, 11093, 11933, 12893, 13397, 19997, 27917

Offset: 1

N. J. A. Sloane, Jan 04 2000

R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A.

R. A. Mollin and H. C. Williams, On a determination of real quadratic fields of class number one and related continued fraction period length less than 25, Proc. Japan Acad. Ser. A Math. Sci. Volume 67, Number 1 (1991), 20-25. (cf. Table 3.1)

Cf. A050950-A050969, A051962-A051965.

A051963 Discriminants of real quadratic fields with class number 1 and related continued fraction period length of 22.

Original entry on oeis.org

166, 489, 491, 523, 643, 662, 947, 971, 1137, 1187, 1427, 1571, 1667, 1713, 1821, 2181, 2217, 2469, 3493, 3693, 3749, 3909, 3947, 4213, 4787, 4989, 5789, 5893, 5909, 6933, 6941, 7509, 7941, 10157, 10533, 10821, 11189, 11469, 12477, 12533, 13733, 14333, 14853, 15069, 15637, 15893, 17813, 19613, 20429, 21117, 23093, 30533, 35237, 36893

Offset: 1

N. J. A. Sloane, Jan 04 2000

R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A.

R. A. Mollin and H. C. Williams, On a determination of real quadratic fields of class number one and related continued fraction period length less than 25, Proc. Japan Acad. Ser. A Math. Sci. Volume 67, Number 1 (1991), 20-25. (cf. Table 3.1)

Cf. A050950-A050969, A051962-A051965.

a(41) and following from Georg Fischer, Sep 20 2021

A051964 Discriminants of real quadratic fields with class number 1 and related continued fraction period length of 23.

Original entry on oeis.org

433, 457, 641, 881, 1381, 1913, 2393, 2749, 3389, 3733, 4421, 5653, 6701, 7349, 7949, 8669, 10253, 11813, 12413, 13709, 13757, 14717, 14813, 14957, 15749, 16229, 16453, 19037, 19421, 22613, 22853, 24317, 27653, 28517, 30197, 31253, 33893, 37397

Offset: 1

N. J. A. Sloane, Jan 04 2000

R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A.

R. A. Mollin and H. C. Williams, On a determination of real quadratic fields of class number one and related continued fraction period length less than 25, Proc. Japan Acad. Ser. A Math. Sci. Volume 67, Number 1 (1991), 20-25. (cf. Table 3.1)

Cf. A050950-A050969, A051962-A051965.

a(37), a(38) from Georg Fischer, Sep 20 2021

A347300 Under Hypothesis(H) (a version of the Generalized Riemann Hypothesis) the only principal real quadratic fields with discriminant d = r^2 + 1 == 1 (mod 4) are the fields Q(sqrt(t)) where t is one of the six numbers listed here.

Original entry on oeis.org

5, 17, 37, 101, 197, 677

Offset: 1

N. J. A. Sloane, Aug 28 2021

Gilles Lachaud, On real quadratic fields, Bull. Amer. Math. Soc., 17:2 (1987), 307-311. See Theorem 3.2.

Cf. A003173, A003174, A014602, A050950.

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

Original entry on oeis.org

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

A050967 Discriminants of real quadratic fields with class number 1 and related continued fraction period length of 18.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A050968 Discriminants of real quadratic fields with class number 1 and related continued fraction period length of 19.

Views

Author

Keywords

References

Links

Crossrefs

A051963 Discriminants of real quadratic fields with class number 1 and related continued fraction period length of 22.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A051964 Discriminants of real quadratic fields with class number 1 and related continued fraction period length of 23.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A347300 Under Hypothesis(H) (a version of the Generalized Riemann Hypothesis) the only principal real quadratic fields with discriminant d = r^2 + 1 == 1 (mod 4) are the fields Q(sqrt(t)) where t is one of the six numbers listed here.

Views

Author

Keywords

References

Crossrefs

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.