A003653 -id:A003653 - cp's OEIS frontend

A226696 Discriminants D of indefinite binary quadratic forms (given in A079896) which allow a solution of the Pell equation x^2 - D*y^2 = -4.

5, 8, 13, 17, 20, 29, 37, 40, 41, 52, 53, 61, 65, 68, 73, 85, 89, 97, 101, 104, 109, 113, 116, 125, 137, 145, 148, 149, 157, 164, 173, 181, 185, 193, 197, 200, 212, 229, 232, 233, 241, 244, 257, 260, 265, 269, 277, 281, 292, 293, 296, 313, 317, 325, 328

Offset: 1

Wolfdieter Lang, Jun 21 2013

nonn

The discriminants D = a(n) which are not squarefree (not in A226693), that is a(n) = k^2*D', lead to a Pell equation for D'. For example, a(2) = 8 leads to x^2 - 2*(2*y)^2 = -4. This has only improper positive integer solutions like (x, 2*y) = (2, 2), (14, 10), (82, 58), ... coming from the proper positive integer solutions of X^2 - 2*Y^2 = -1, (X, Y) = (1, 1), (7, 5), (41, 29), ...

The +4 Pell equation has a solution (in fact infinitely many solutions) for each D from A079896.

			Positive fundamental solutions (proper or improper):
n=1,  D=5:  (1, 1), (11, 5); (4, 2)
n=2,  D=8:  (2, 1)
n=3,  D=13: (3, 1), (393, 109); (36, 10)
n=4,  D=17: no proper solution; (8, 2)
n=5,  D=20: (4, 1)
n=6,  D=29: (5, 1), (3775, 701); (140, 26)
n=7,  D=37: no proper solution; (12, 2)
n=8,  D=40: (6, 1)
n=9,  D=41: no proper solution; (64, 10)
n=10, D=52: (36, 5)
n=11, D=53: (7, 1), (18557, 2549); (364, 50)
...

D. A. Buell, Binary Quadratic Forms, Springer, 1989, Sections 3.2 and 3.3, pp. 31-48.
A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, Paragraph 32, pp. 121-126.

Robin Visser, Table of n, a(n) for n = 1..10000

Cf. A079896, A226165, A226693.

A003653 is a subsequence listing the fundamental discriminants in this sequence.

solQ[d_] := Mod[d, 4] <= 1 && !IntegerQ[Sqrt[d]] && Reduce[x^2 - d*y^2 == -4, {x, y}, Integers] =!= False; Select[Range[328], solQ ] (* Jean-François Alcover, Jul 03 2013 *)

isA226696(D) = if(D%4<=1&&!issquare(D), for(n=1,oo,if(issquare(D*n^2-4),return(1));if(issquare(D*n^2+4),return(0))), 0) \\ Jianing Song, Mar 02 2019

The sequence lists the increasing D values which are not a square, are 1 (mod 4) or 0 (mod 4) (members of A079896) and allow a solution (in fact infinitely many solutions) of x^2 - D*y^2 = -4.

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

A226696 Discriminants D of indefinite binary quadratic forms (given in A079896) which allow a solution of the Pell equation x^2 - D*y^2 = -4.

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

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

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