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%I #41 Mar 31 2019 02:18:22
%S 1,3,3,13,5,39,59,7,23,221,59,9,9,477,31,2175,103,8807,41571,8005,13,
%T 2047,2999,127539,527593,15,15,2489,1917,373,340551,11759,9409,4109,
%U 52778687,801,19,137913,113759383,137,16437,12311,21,21,15732537,1275,1729,7204587,305987,67
%N x-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 = a(n)^2 - (-1)^((p+1)/4), Y = a(n)*A306566(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) = (a(n) + A306566(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 a(n) = x and A306566(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, x=1; while(!issquare((x^2 - 2*(-1)^((p+1)/4))/p), x++); x)
%o forprime(p=3, 500, if(p%4==3, print1(b(p), ", ")))
%Y Cf. A002145, A306566 (y-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 1,2
%A _Jianing Song_, Mar 25 2019
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