English: Primes ${\displaystyle \pi =a+b\phi }$ of the real quadratic ring ${\displaystyle 𝒪=\mathbb{Z}[\phi ]}$ embedded into a point-lattice ${\displaystyle \Lambda }$ based on unit vectors ${\displaystyle (1,1)\text{ and }(1-\phi ,\phi ),}$ with golden ratio ${\displaystyle \phi =\frac{1}{2}(1+\sqrt{5}).}$ ${\displaystyle |x|,|y|<40,}$ with ${\displaystyle x=\text{ conjugate }(a+b)-b\phi ,}$ and ${\displaystyle y=a+b\phi .}$ 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 ${\displaystyle \pm {\phi }^n\pi \text{ with }n\ne 0.}$ The fundamental parallelogram with area ${\displaystyle \delta (\Lambda )=\sqrt{5}}$ in gold, the hyperbolas pass through the factors ${\displaystyle \pm \{(2+\phi ),(3-\phi )\}}$ and ${\displaystyle \pm (1-2\phi )}$ 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.