A226165 -id:A226165 - cp's OEIS frontend

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

Wolfdieter Lang, Jun 21 2013

nonn

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.

			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)
...

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.

Robin Visser, Table of n, a(n) for n = 1..10000

Cf. A079896, A226165, A226693.

A003653 is a subsequence listing the fundamental discriminants in this sequence.

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 *)

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

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.

A226693 Squarefree parts of A079896(n), n>= 1.

Original entry on oeis.org

5, 2, 3, 13, 17, 5, 21, 6, 7, 29, 2, 33, 37, 10, 41, 11, 5, 3, 13, 53, 14, 57, 15, 61, 65, 17, 69, 2, 73, 19, 77, 5, 21, 85, 22, 89, 23, 93, 6, 97, 101, 26, 105, 3, 109, 7, 113, 29, 13, 30, 31, 5, 2, 129, 33, 133, 34, 137, 35, 141, 145, 37, 149, 38, 17, 39, 157, 10

Offset: 1

Wolfdieter Lang, Jun 15 2013

a(n) is the squarefree part of the discriminant D(n) = A079896(n) of indefinite binary quadratic forms. Certain quadratic irrationals, called omega_p(D(n)), related to the principal indefinite form of discriminant D(n) are integers in the quadratic number field Q(sqrt(a(n))). See A226166 for the definition of these irrationals omega_p(D(n)) using the D. A. Buell reference, p. 31 and p. 26.

For discriminants D == 1 (mod 4) these squarefree parts are given in A226165. For D == 0 (mod 4) the squarefree parts are given in A002734 corresponding to A000037 = D/4.

D. A. Buell, Binary Quadratic Forms, Springer, 1989.

Gheorghe Coserea, Table of n, a(n) for n = 1..20001

Cf. A079896, A226165, A002734.

SquareFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n]); SquareFreePart /@ Select[ Range[160], ! IntegerQ[Sqrt[#]] && Mod[#, 4] < 2 &] (* Jean-François Alcover, Jun 25 2013 *)

A079896_list(N) = {
  my(n = 1, v = vector(N), top = 0);
  while (top < N, if (n%4 < 2 && !issquare(n), v[top++] = n); n++;);
  return(v);
};
apply(core, A079896_list(68)) \\ Gheorghe Coserea, Nov 10 2016

a(n) = squarefree part of D(n) = A079896(n), n >= 1, the numbers 0 and 1 (mod 4), not a square.

Offset corrected by Robin Visser, Jun 01 2025

cp's OEIS Frontend

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.

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