Constructing $x^2$ for primes $p=a{x}^2+b{y}^2$

Abstract

Let $a$ and $b$ be positive integers and let $p$ be an odd prime such that $p=a{x}^2+b{y}^2$ 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 $(-1{)}^{\frac{a+b}{2}x+\frac{b+1}{2}}(4a{x}^2-2p)=\lambda (a,b;((ab+1)p-a-b)/8+1)$. We also give formulas for $\lambda (1,3;n+1),\lambda (1,7;2n+1)$, $\lambda (3,5;2n+1)$ and $\lambda (1,15;4n+1)$.

16 pages