Constructing $x^2$ for primes $p=ax^2+by^2$

Abstract

Let $a$ and $b$ be positive integers and let $p$ be an odd prime such that $p=ax^2+by^2$ for some integers $x$ and $y$. Let $\lambda(a,b;n)$ be given by $q\prod_{k=1}^\infty (1-q^{ak})^3(1-q^{bk})^3 = \sum_{n=1}^\infty \lambda(a,b;n)q^n$. In the paper, using Jacobi's identity $\prod_{n=1}^\infty (1-q^n)^3 = \sum_{k=0}^\infty (-1)^k(2k+1)q^{\frac{k(k+1)}2}$ we construct $x^2$ in terms of $\lambda(a,b;n)$. For example, if $2\nmid ab$ and $p\nmid ab(ab+1)$, then $(-1)^{\frac{a+b}2x+\frac{b+1}2}(4ax^2-2p) = \lambda(a,b;((ab+1)p-a-b)/8+1)$. We also give formulas for $\lambda(1,3;n+1),\lambda(1,7;2n+1)$, $\lambda(3,5;2n+1)$ and $\lambda(1,15;4n+1)$.