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A226696
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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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2
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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
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1,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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)
...
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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A003653 is a subsequence listing the fundamental discriminants in this sequence.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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