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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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, Einfuehrung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, Paragraph 32, p. 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-Fran├žois Alcover, Jul 03 2013 *)

CROSSREFS

Cf. A079896, A226165, A226693.

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

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Last modified December 15 20:10 EST 2018. Contains 318154 sequences. (Running on oeis4.)