%I #34 Mar 31 2019 02:18:36

%S 1,1,1,3,1,7,9,1,3,27,7,1,1,47,3,193,9,747,3383,627,1,153,217,9041,

%T 36321,1,1,161,121,23,20687,699,537,233,2900979,43,1,7199,5843427,7,

%U 803,593,1,1,731153,59,79,326471,13809,3,7,12507,541137,11,563210019

%N y-value of the smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4), p = A002145(n).

%C a(n) exists for all n.

%C X = A306529(n)^2 - (-1)^((p+1)/4), Y = A306529(n)*a(n) gives the smallest solution to x^2 - p*y^2 = 1, p = A002145(n). As a result, all the positive solutions to x^2 - p*y^2 = 2*(-1)^((p+1)/4) are given by (x_n, y_n) where x_n + (y_n)*sqrt(p) = (A306529(n) + a(n)*sqrt(p))*(X + Y*sqrt(p))^n.

%F If the continued fraction of sqrt(A002145(n)) is [a_0; {a_1, a_2, ..., a_(k-1), a_k, a_(k-1), ..., a_1, 2*a_0}], where {} is the periodic part, let x/y = [a_0; a_1, a_2, ..., a_(k-1)], gcd(x, y) = 1, then A306529(n) = x and a(n) = y.

%e The smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4) for the first primes congruent to 3 modulo 4:

%e n | Equation | x_min | y_min

%e 1 | x^2 - 3*y^2 = -2 | 1 | 1

%e 2 | x^2 - 7*y^2 = +2 | 3 | 1

%e 3 | x^2 - 11*y^2 = -2 | 3 | 1

%e 4 | x^2 - 19*y^2 = -2 | 13 | 3

%e 5 | x^2 - 23*y^2 = +2 | 5 | 1

%e 6 | x^2 - 31*y^2 = +2 | 39 | 7

%e 7 | x^2 - 43*y^2 = -2 | 59 | 9

%e 8 | x^2 - 47*y^2 = +2 | 7 | 1

%e 9 | x^2 - 59*y^2 = -2 | 23 | 3

%o (PARI) b(p) = if(isprime(p)&&p%4==3, y=1; while(!issquare(p*y^2 + 2*(-1)^((p+1)/4)), y++); y)

%o forprime(p=3, 500, if(p%4==3, print1(b(p), ", ")))

%Y Cf. A002145, A306529 (x-values).

%Y Similar sequences: A094048, A094049 (x^2 - A002144(n)*y^2 = -1); A306618, A306619 (2*x^2 - A002145(n)*y^2 = (-1)^((p+1)/4))).

%K nonn

%O 1,4

%A _Jianing Song_, Mar 25 2019