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.
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
Offset: 1
Keywords
Examples
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) ...
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.
Links
- Robin Visser, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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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 *) -
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
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.
Comments