We know that the normalisation of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt{d})$ where $d\in \mathbb{Z}$ is
$$O=\mathbb{Z}[\beta], \text{$\beta=\sqrt{d}$ if $d\equiv2,3 \pmod 4$}; \ \frac{1+\sqrt{d}}{2} \text{if} \ d\equiv1 \pmod 4. $$
How to show $$O \ \text{is UFD (or PID)} \ \text{if and only if} \ d=-1,-2,-3,-7,-11,-19,-43,-67,-163 $$
This was first proved in $1967$ in
1.) H. M. Stark, A complete determination of the complex quadratic fields of class number one, Michigan Mathematics Journal (1967), 1-27.
2.) Alan Baker, Linear forms in logarithms of algebraic numbers, Mathematika (1966), 204-216.
There is an earlier "proof" of Heegner in $1952$, which has some gaps, but is essentially correct. A further new proof was given by Jean-Pierre Serre, in
3.) Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, ch. Appendix: The Class Number 1 Problem and Integral Points on Modular Curves, Vieweg, 1989.