# The least prime number represented by a binary quadratic form

### Naser Talebizadeh Sardari

Max-Planck-Institut für Mathematik, Bonn, Germany

## Abstract

Let $D < 0$ be a fundamental discriminant and $h(D)$ be the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{D})$. Let $R(X,D)$ be the number of the classes of the binary quadratic forms of discriminant $D$ which represent a prime number in the interval [$X, 2X$]. Moreover, assume that $\pi_D(X)$ is the number of primes which split in $\mathbb{Q}(\sqrt{D})$ with norm in the interval [$X, 2X$]. We prove that

$\Big(\frac{\pi_D(X)}{\pi(X)}\Big)^2 \ll \frac{R(X,D)}{h(D)}\Big(1+\frac{h(D)}{\pi(X)}\Big),$

where $\pi(X)$ is the number of the primes is the number of primes in the interval [$X, 2X$] and the implicit constant in $\ll$ is independent of $D$ and $X$.

## Cite this article

Naser Talebizadeh Sardari, The least prime number represented by a binary quadratic form. J. Eur. Math. Soc. 23 (2021), no. 4, pp. 1161–1223

DOI 10.4171/JEMS/1031