A Necessary and Sufficient Condition for k = ℚ 4 n 2 + 1 to Have Class Number ω n + c

Abstract

In this paper, we give a necessary and sufficient condition for real quadratic field $k=\mathrm{\mathbb{Q}}\left(\sqrt{4n{}^2+1}\right)$ to have class number $\omega \left(n\right)+c$ where $c\in \left\{1,2\right\}$ and $\omega \left(n\right)$ is the number of odd prime divisors of $n$ .