A306638 a(n) is the norm of the fundamental unit of binary quadratic forms with discriminant D = A079896(n).

-1, -1, 1, -1, -1, -1, 1, 1, 1, -1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, -1, 1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, -1, -1, 1, 1, -1, 1, -1, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, -1, -1

Offset: 1

Author

Jianing Song, Mar 02 2019

Keywords

• sign

Comments

The fundamental unit of binary quadratic forms with discriminant D is the number (x_1 + (y_1)*sqrt(D))/2, where (x_1,y_1) is the smallest solution to x^2 - D*y^2 = +-4. Each term is either -1 or 1 depending on whether (x_1)^2 - D*(y_1)^2 = -4 or 4.

All solutions to x^2 - D*y^2 = +-4 are given by the identity (x_n + (y_n)*sqrt(D))/2 = ((x_1 + (y_1)*sqrt(D))/2)^n.

The discriminants D corresponding to (x_1)^2 - D*(y_1)^2 = -4 are listed in A226696.

Examples

			Fundamental units and their norms for the first 15 discriminants in the form (X + Y*sqrt(D))/2 (N = (X^2 - D*Y^2)/4 are the corresponding norms) are:
   D |  X |  Y |  N
   5 |  1 |  1 | -1
   8 |  2 |  1 | -1
  12 |  4 |  1 |  1
  13 |  3 |  1 | -1
  17 |  8 |  2 | -1
  20 |  4 |  1 | -1
  21 |  5 |  1 |  1
  24 | 10 |  2 |  1
  28 | 16 |  3 |  1
  29 |  5 |  1 | -1
  32 |  6 |  1 |  1
  33 | 46 |  8 |  1
  37 | 12 |  2 | -1
  40 |  6 |  1 | -1
  41 | 64 | 10 | -1
		

References

• D. A. Buell, Binary Quadratic Forms, Springer, 1989, Sections 3.2 and 3.3, pp. 31-48.

Links

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

Crossrefs

Cf. A079896, A226696.

A014077 is a subsequence listing the corresponding values for only fundamental discriminants (A003658).

Programs

Julia

using Nemo
function b(D)
    for j in 1:10000
        issquare(D*j^2 - 4) && return -1
        issquare(D*j^2 + 4) && return 1
    end
0 end
F = findall(n -> ZZ(n) % 4 <= 1 && !issquare(ZZ(n)), 1:100)
map(n -> b(ZZ(n)), F) |> println # Peter Luschny, Mar 08 2019

PARI

b(D) = for(n=1, oo, if(issquare(D*n^2-4), return(-1)); if(issquare(D*n^2+4), return(1)))
for(n=2, 200, if(n%4 <= 1 && !issquare(n), print1(b(n), ", ")))

Formula

a(n) = -1 if D = A079896(n) is in A226696, otherwise 1.

Extensions

Offset changed to 1 by Robin Visser, Jun 09 2025