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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.

%I A226696 #29 Jun 10 2025 21:48:27
%S A226696 5,8,13,17,20,29,37,40,41,52,53,61,65,68,73,85,89,97,101,104,109,113,
%T A226696 116,125,137,145,148,149,157,164,173,181,185,193,197,200,212,229,232,
%U A226696 233,241,244,257,260,265,269,277,281,292,293,296,313,317,325,328
%N 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.
%C A226696 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), ...
%C A226696 The +4 Pell equation has a solution (in fact infinitely many solutions) for each D from A079896.
%D A226696 D. A. Buell, Binary Quadratic Forms, Springer, 1989, Sections 3.2 and 3.3, pp. 31-48.
%D A226696 A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, Paragraph 32, pp. 121-126.
%H A226696 Robin Visser, <a href="/A226696/b226696.txt">Table of n, a(n) for n = 1..10000</a>
%F A226696 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.
%e A226696 Positive fundamental solutions (proper or improper):
%e A226696 n=1,  D=5:  (1, 1), (11, 5); (4, 2)
%e A226696 n=2,  D=8:  (2, 1)
%e A226696 n=3,  D=13: (3, 1), (393, 109); (36, 10)
%e A226696 n=4,  D=17: no proper solution; (8, 2)
%e A226696 n=5,  D=20: (4, 1)
%e A226696 n=6,  D=29: (5, 1), (3775, 701); (140, 26)
%e A226696 n=7,  D=37: no proper solution; (12, 2)
%e A226696 n=8,  D=40: (6, 1)
%e A226696 n=9,  D=41: no proper solution; (64, 10)
%e A226696 n=10, D=52: (36, 5)
%e A226696 n=11, D=53: (7, 1), (18557, 2549); (364, 50)
%e A226696 ...
%t A226696 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 *)
%o A226696 (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
%Y A226696 Cf. A079896, A226165, A226693.
%Y A226696 A003653 is a subsequence listing the fundamental discriminants in this sequence.
%K A226696 nonn
%O A226696 1,1
%A A226696 _Wolfdieter Lang_, Jun 21 2013