Constructing x2 for primes p=ax2+by2

Let a and b be positive integers and let p be an odd prime such that p=ax2+by2 for some integers x and y. Let λ(a,b;n) be given by . In this paper, using Jacobiʼs identity , we construct x2 in terms of λ(a,b;n). For example, if 2∤ab and p∤ab(ab+1), then . We also give formulas for λ(1,3;n+1),λ(1,7;2n+1), λ(3,5;2n+1) and λ(1,15;4n+1).