English: Primes ${\displaystyle \pi =a+b\varphi }$ of the real quadratic ring ${\displaystyle {\mathcal {O}}=\mathbb {Z} [\varphi ]}$ embedded into a point-lattice ${\displaystyle \Lambda }$ based on unit vectors ${\displaystyle \left(1,1\right){\text{ and }}\left(1-\varphi ,\,\varphi \right),}$ with golden ratio ${\displaystyle \varphi ={\tfrac {1}{2}}(1+{\sqrt {5}}).}$ ${\displaystyle \,\vert x\vert ,\vert y\vert <40,}$ with ${\displaystyle x={\text{ conjugate }}(a+b)-b\varphi \,,}$ and ${\displaystyle y=a+b\varphi .}$ Primes with positive or negative norm ${\displaystyle N\pi =a^{2}+ab-b^{2}}$ in successively green and red. Only primes of minimal trace ${\displaystyle 2a+b}$ are plotted, leaving out associates ${\displaystyle \pm \varphi ^{n}\pi {\text{ with }}n\neq 0.}$ The fundamental parallelogram with area ${\displaystyle \delta (\Lambda )={\sqrt {5}}}$ in gold, the hyperbolas pass through the factors ${\displaystyle \pm \left\{\left(2+\varphi \right),\left(3-\varphi \right)\right\}}$ and ${\displaystyle \pm \left(1-2\varphi \right)}$ of ramified prime ${\displaystyle 5.}$^[1]

1. ↑ Dekker, T. J. (2010) Primes in quadratic fields, DOI:10.48550/arXiv.1001.5214, page 13, Appendix 2

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  GoldenLattice.svg Close-up of the centre region.
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  SilverPrimes.svg Matching silver ratio diagram.