

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
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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. 3148.
A. Scholz and B. Schoeneberg, Einfuehrung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, Paragraph 32, p. 121126.


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 ] (* JeanFrançois 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



