Solving x^2+n*y^2=a without factoring positive $n$?

thue() appears to solve x^2+n*y^2=a without factoring
positive $n$ when we declare n as prime via addprimes():

? p=nextprime(2*10^8);q=nextprime(3*p);n=p*q;a=(p-1)^2+n*200^2;addprimes([n]);K=
thue(thueinit(x^2+n,0),a)

Given n and factored a, what is the complexity of solving the
equation? (the solution is bounded by sqrt(a))