cp's OEIS Frontend

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

%I A306638 #35 Jun 11 2025 14:25:06
%S A306638 -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,
%T A306638 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,
%U A306638 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
%N A306638 a(n) is the norm of the fundamental unit of binary quadratic forms with discriminant D = A079896(n).
%C A306638 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.
%C A306638 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.
%C A306638 The discriminants D corresponding to (x_1)^2 - D*(y_1)^2 = -4 are listed in A226696.
%D A306638 D. A. Buell, Binary Quadratic Forms, Springer, 1989, Sections 3.2 and 3.3, pp. 31-48.
%H A306638 Robin Visser, <a href="/A306638/b306638.txt">Table of n, a(n) for n = 1..10000</a>
%F A306638 a(n) = -1 if D = A079896(n) is in A226696, otherwise 1.
%e A306638 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:
%e A306638    D |  X |  Y |  N
%e A306638    5 |  1 |  1 | -1
%e A306638    8 |  2 |  1 | -1
%e A306638   12 |  4 |  1 |  1
%e A306638   13 |  3 |  1 | -1
%e A306638   17 |  8 |  2 | -1
%e A306638   20 |  4 |  1 | -1
%e A306638   21 |  5 |  1 |  1
%e A306638   24 | 10 |  2 |  1
%e A306638   28 | 16 |  3 |  1
%e A306638   29 |  5 |  1 | -1
%e A306638   32 |  6 |  1 |  1
%e A306638   33 | 46 |  8 |  1
%e A306638   37 | 12 |  2 | -1
%e A306638   40 |  6 |  1 | -1
%e A306638   41 | 64 | 10 | -1
%o A306638 (PARI) b(D) = for(n=1, oo, if(issquare(D*n^2-4), return(-1)); if(issquare(D*n^2+4), return(1)))
%o A306638 for(n=2, 200, if(n%4 <= 1 && !issquare(n), print1(b(n), ", ")))
%o A306638 (Julia) using Nemo
%o A306638 function b(D)
%o A306638     for j in 1:10000
%o A306638         issquare(D*j^2 - 4) && return -1
%o A306638         issquare(D*j^2 + 4) && return 1
%o A306638     end
%o A306638 0 end
%o A306638 F = findall(n -> ZZ(n) % 4 <= 1 && !issquare(ZZ(n)), 1:100)
%o A306638 map(n -> b(ZZ(n)), F) |> println # _Peter Luschny_, Mar 08 2019
%Y A306638 Cf. A079896, A226696.
%Y A306638 A014077 is a subsequence listing the corresponding values for only fundamental discriminants (A003658).
%K A306638 sign
%O A306638 1,1
%A A306638 _Jianing Song_, Mar 02 2019
%E A306638 Offset changed to 1 by _Robin Visser_, Jun 09 2025