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.
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
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Jun 21 2013
STATUS
approved