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

1,1

Comments

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.

References

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.

Links

Table of n, a(n) for n=1..55.

Formula

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.

Example

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)
...

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

Cf. A079896, A226165, A226693.

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

Sequence in context: A314425 A058240 A097268 * A285973 A219639 A314426

Adjacent sequences: A226693 A226694 A226695 * A226697 A226698 A226699

Keyword

nonn

Author

Wolfdieter Lang, Jun 21 2013

Status

approved