A003656 -id:A003656 - cp's OEIS frontend

A003246 Discriminants of real quadratic norm-Euclidean fields (a finite sequence).

5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 41, 44, 57, 73, 76

Offset: 1

Euclidean fields that are not norm-Euclidean, such as Q(sqrt(14)) and Q(sqrt(69)), are not included. Actually, assuming GCH, a real quadratic field is Euclidean if and only if it is a PID (equivalently, if and only if it is a UFD). - Jianing Song, Jun 09 2022

W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 2, p. 57.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 294.

S. R. Finch, Class number theory [Cached copy, with permission of the author]
Erich Kaltofen and Heinrich Rolletschek, Computing greatest common divisors and factorizations in quadratic number fields, Mathematics of Computation 53.188 (1989): 697-720. See page 698.
A. M. Odlyzko, Letters to N. J. A. Sloane Feb 1974
P. Samuel, Unique factorization, Amer. Math. Monthly 75 (1968), 945-952.
Peter J. Weinberger, On Euclidean rings of algebraic integers, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 321-332.
Index entries for sequences related to quadratic fields

Cf. A003174, A003656.

A003174 = {2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73}; Sort[ NumberFieldDiscriminant /@ Sqrt[A003174]] (* Jean-François Alcover, Jul 18 2012 *)

for(n=1,99,is_A003174(n) && print1(quaddisc(n)",")) \\ M. F. Hasler, Jan 26 2014

Equals A037449(A003174) as a set, not composition of functions (values are sorted by size; it turns out that a(n) is different from A037449(A003174(n)) for all n=1,...,16). - M. F. Hasler, Jan 26 2014

A052475 Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(5)).

Original entry on oeis.org

40, 60, 65, 85, 105, 120, 140, 165, 185, 205, 220, 265, 280, 285, 305, 345, 365, 380, 385, 440, 460, 465, 485, 545, 565, 620, 645, 665, 685, 705, 745, 760, 805, 860, 865, 885, 920, 965, 1005, 1065, 1085, 1165, 1180, 1185, 1205, 1240, 1245, 1265, 1285

Offset: 1

N. J. A. Sloane, Mar 15 2000

H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, pp. 534-535.

Henri Cohen and X.-F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244.

Cf. A035120, A003656, A052476, A052477, A052478, A052479.

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 15 2002

A094613 Fundamental discriminants of real quadratic number fields with class number 4.

Original entry on oeis.org

145, 328, 445, 505, 520, 680, 689, 777, 780, 793, 840, 876, 897, 901, 905, 924, 1020, 1045, 1096, 1105, 1145, 1160, 1164, 1221, 1288, 1292, 1313, 1320, 1365, 1480, 1560, 1640, 1677, 1736, 1740, 1745, 1752, 1820, 1848, 1885, 1932, 2005, 2040, 2056, 2120, 2145

Offset: 1

Eric W. Weisstein, May 14 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612, A094614, A218156-A218160.

Select[Range[2145], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 4 &] (* Arkadiusz Wesolowski, Oct 22 2012 *)

{ok(n) = n>10 && isfundamental(n) && qfbclassno(n)==4};
for(n=1, 2500, if(ok(n)==1, print1(n, ", "))) \\ G. C. Greubel, Mar 01 2019

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 4;
A094613 = lambda n: filter(is_fund_and_qfbcn_1, (1, 2, .., n));
A094613(2500) # G. C. Greubel, Mar 01 2019

A342368 Fundamental discriminants of real quadratic number fields with odd class number > 1.

Original entry on oeis.org

229, 257, 316, 321, 401, 469, 473, 568, 577, 733, 761, 817, 892, 993, 1009, 1016, 1093, 1101, 1129, 1229, 1257, 1297, 1304, 1373, 1393, 1429, 1436, 1489, 1509, 1601, 1641, 1756, 1761, 1772, 1897, 1901, 1929, 1957, 1996, 2021, 2029, 2081, 2089, 2101, 2153, 2177, 2213

Offset: 1

Jianing Song, Mar 09 2021

nonn

For a positive fundamental discriminant d, the class number of the real quadratic field of discriminant d is odd if and only if d = 8 or is of one of the three following forms: (i) p, where p is a prime congruent to 1 modulo 4; (ii) 4p or 8p, where p is a prime congruent to 3 modulo 4; (iii) pq, where p, q are distinct primes congruent to 3 modulo 4. See Theorem 1 and Theorem 2 of Ezra Brown's link. A003656 gives the case where the class number is 1.

			The class number of the quadratic field with discriminant 229 (namely Q(sqrt(229))) is 3, so 229 is a term.
The class number of the quadratic field with discriminant 1756 (namely Q(sqrt(439))) is 5, so 1756 is a term.

Jianing Song, Table of n, a(n) for n = 1..22463 (all terms <= 10^6).
Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.
Henri Cohen and X.-F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244.
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656.

isA342368(D) = if((D>1) && isfundamental(D), my(h=quadclassunit(D)[1]); (h%2)&&(h>1), 0)

A003655 Discriminants of real quadratic fields with narrow class number 1.

Original entry on oeis.org

5, 8, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 233, 241, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 593, 601, 613, 617, 641, 653, 661, 673

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

Or, positive fundamental discriminants with form class number 1.

All terms except 8 are primes congruent to 1 modulo 4. - Jianing Song, Jul 20 2022

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.
Charles Delorme and Guillermo Pineda-Villavicencio, Quadratic Form Representations via Generalized Continuants, Journal of Integer Sequences, Vol. 18 (2015), Article 15.6.4.

Equals {8} U (A003656 intersect A002144).

Equals A003656 \ A327297.

isA003655(n) = (n==8) || (isprime(n) && (n%4==1) && (qfbclassno(n)==1)) \\ Jianing Song, Jul 20 2022

Better definition from David Brink, Dec 30 2007, Jan 01 2008

A218157 Fundamental discriminants of real quadratic number fields with class number 7.

Original entry on oeis.org

577, 1009, 1601, 1761, 2029, 2913, 4229, 4348, 5176, 5273, 5417, 7736, 8097, 8661, 8773, 9004, 9029, 9049, 9101, 9208, 9289, 9868, 10117, 10313, 10357, 10713, 10957, 11021, 11053, 11269, 11537, 11621, 12497, 12977, 13049, 13313, 13701, 14201, 15277, 15809

Offset: 1

Arkadiusz Wesolowski, Oct 22 2012

nonn

Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612-A094614, A218156, A218158-A218160.

Select[Range[15809], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 7 &]

A218158 Fundamental discriminants of real quadratic number fields with class number 8.

Original entry on oeis.org

904, 1596, 1705, 1768, 1785, 2584, 2605, 2705, 3081, 3196, 3201, 3480, 3640, 3976, 4092, 4161, 4305, 4488, 4620, 4669, 4956, 5160, 5196, 5249, 5305, 5404, 5513, 5713, 5772, 5784, 5865, 6360, 6409, 6565, 6757, 6953, 6972, 7449, 7585, 7656, 7788, 7833, 7980, 8005

Offset: 1

Arkadiusz Wesolowski, Oct 22 2012

nonn

Victor Y. Wang, On Hilbert 2-class fields and 2-towers of imaginary quadratic number fields, arXiv preprint arXiv:1508.06552, 2015
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612-A094614, A218156-A218157, A218159-A218160.

Select[Range[8005], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 8 &]

A218159 Fundamental discriminants of real quadratic number fields with class number 9.

Original entry on oeis.org

1129, 3137, 4409, 5521, 6616, 6809, 7573, 7873, 10273, 10721, 11641, 12409, 12657, 13069, 14876, 15629, 16321, 17273, 17989, 18136, 18633, 19441, 21781, 22492, 22497, 23512, 24029, 24169, 24697, 24781, 25361, 26573, 27221, 27349, 28901, 29317, 31897

Offset: 1

Arkadiusz Wesolowski, Oct 22 2012

nonn

Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003656, A094619, A094612-A094614, A218156-A218158, A218160.

Select[Range[31897], NumberFieldDiscriminant@Sqrt[#] == # && NumberFieldClassNumber@Sqrt[#] == 9 &]

A349649 Discriminants of real quadratic fields whose fundamental unit has norm 1.

Original entry on oeis.org

12, 21, 24, 28, 33, 44, 56, 57, 60, 69, 76, 77, 88, 92, 93, 105, 120, 124, 129, 133, 136, 140, 141, 152, 156, 161, 165, 168, 172, 177, 184, 188, 201, 204, 205, 209, 213, 217, 220, 221, 236, 237, 248, 249, 253, 264, 268, 273, 280, 284, 285, 301, 305, 309, 312, 316

Offset: 1

Arkadiusz Wesolowski, Nov 23 2021

nonn

D. A. Buell, Binary Quadratic Forms, Springer-Verlag, NY, 1989, pp. 92-93.
Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, NY, 2000, p. 151.

Index entries for sequences related to quadratic fields

Cf. A003658, A003653 (discriminants of real quadratic fields whose fundamental unit has norm -1).

Cf. A003656, A327297 (a subsequence).

isok(D) = isfundamental(D) && norm(quadunit(D))==1;

A350165 Fundamental discriminants of real quadratic number fields with odd class number > 1 whose fundamental unit has norm -1.

Original entry on oeis.org

229, 257, 401, 577, 733, 761, 1009, 1093, 1129, 1229, 1297, 1373, 1429, 1489, 1601, 1901, 2029, 2081, 2089, 2153, 2213, 2557, 2677, 2713, 2777, 2857, 2917, 3121, 3137, 3181, 3221, 3229, 3253, 3877, 3889, 4001, 4229, 4357, 4409, 4441, 4481, 4493, 4597, 4649, 4729, 4889, 4933

Offset: 1

Jianing Song, Dec 29 2021

nonn

Prime terms of A342368.

For a positive fundamental discriminant d, the class number of the real quadratic field of discriminant d is odd if and only if d = 8 or is of one of the three following forms: (i) p, where p is a prime congruent to 1 modulo 4; (ii) 4p or 8p, where p is a prime congruent to 3 modulo 4; (iii) pq, where p, q are distinct primes congruent to 3 modulo 4. See Theorem 1 and Theorem 2 of Ezra Brown's link. This sequence gives values for d in the case (i) and that the real quadratic number field with discriminant d has odd class number > 1.

			229 is a term since the quadratic field with discriminant 229 (Q(sqrt(229))) has class number 5. The fundamental unit of that field ((15+sqrt(229))/2) has norm -1.
401 is a term since the quadratic field with discriminant 401 (Q(sqrt(401))) has class number 5. The fundamental unit of that field (20+sqrt(401)) has norm -1.

Winston de Greef, Table of n, a(n) for n = 1..10000
Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.
Henri Cohen and X.-F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244.
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Intersection of A342368 and A003653. Equals A342368 \ A349419.

Cf. A003658, A003656.

isA350165(D) = if(isprime(D) && isfundamental(D), my(h=quadclassunit(D)[1]); (h%2)&&(h>1), 0)

cp's OEIS Frontend

A003246 Discriminants of real quadratic norm-Euclidean fields (a finite sequence).

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A052475 Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(5)).

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A094613 Fundamental discriminants of real quadratic number fields with class number 4.

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A342368 Fundamental discriminants of real quadratic number fields with odd class number > 1.

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A003655 Discriminants of real quadratic fields with narrow class number 1.

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A218157 Fundamental discriminants of real quadratic number fields with class number 7.

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A218158 Fundamental discriminants of real quadratic number fields with class number 8.

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A218159 Fundamental discriminants of real quadratic number fields with class number 9.

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A349649 Discriminants of real quadratic fields whose fundamental unit has norm 1.

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