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A306638 a(n) is the norm of the fundamental unit of binary quadratic forms with discriminant D = A079896(n). 0
-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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=0..85.

FORMULA

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

EXAMPLE

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

PROG

(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), ", ")))

(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

CROSSREFS

Cf. A079896, A226696.

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

Sequence in context: A098417 A143622 A246016 * A076479 A155040 A209661

Adjacent sequences:  A306635 A306636 A306637 * A306639 A306640 A306641

KEYWORD

sign

AUTHOR

Jianing Song, Mar 02 2019

STATUS

approved

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Last modified November 17 11:02 EST 2019. Contains 329226 sequences. (Running on oeis4.)