Consider the Euclidean complex quadratic rings.
There are only a $5$ of them.
$$ \Bbb Z[\sqrt{-1}],\Bbb Z[\sqrt {-2}], \Bbb Z[\frac{1+\sqrt {-3}}{2}], \Bbb Z[\frac{1+\sqrt {-7}}{2}], \Bbb Z[\frac{1+\sqrt {-11}}{2}]$$
Some of them are named and heavily researched such as
$$ \Bbb Z[\sqrt{-1}], \Bbb Z[\frac{1+\sqrt {-3}}{2}], \Bbb Z[\frac{1+\sqrt {-7}}{2}]$$
Known as the Gaussian integers, Eisenstein integers and Kleinian integers.
(apparantly Kleinians are popular for cryptography, but I am unsure why ?)
Ed Pegg has named
$$ \Bbb Z[\sqrt {-2}]$$
The Hippasus integers and the related primes, the Hippasus Primes.
See for instance here :
https://community.wolfram.com/groups/-/m/t/965609
and here :
I am uncertain if
$$\Bbb Z[\frac{1+\sqrt {-11}}{2}]$$
Is named or investigated alot.
Anyway A lot of questions pop up for the primes in those rings.
And questions has already been asked about them.
Although they are pretty well understood with quadratic residues and primes of the forms $u \mod v$ etc, combined ideas and visuals are still mysterious.
For instance :
Here I ask for the prime distribution of those two rings within a radius compared to eachother.
But Now I want to focus more on the visualizations that Ed plotted and related.
In fact instead of thinking about primes in one of those rings, I wondered about primes in all of those or some of those rings simultanously, meaning stuff like :
CASE A :
$$a + b(\frac{1 + \sqrt{-7}}{2}),a + b(\frac{1 + \sqrt{-11}}{2})$$
are both primes for the same values $a,b$.
Similar to this related post :
List of prime numbers in imaginary quadratic fields with UFD
where they stick to the primes that are real integers.
I wonder if the visuals( Like Ed made) for "simultanously prime " like in CASE 1 or similar has any remarkable patterns.
And I wonder if those remarkable patterns in the visuals remain for larger data (aka zooming out). (I also wonder if zooming out for the original visuals in Ed's link maintain fascinating for that matter)
It might be interesting to note the norms of those rings;
$$ \Bbb Z[\sqrt{-1}],\Bbb Z[\sqrt {-2}]$$
have norms $a^2 + b^2$ and $a^2 + 2 b^2$ whereas
$$\Bbb Z[\frac{1+\sqrt {-3}}{2}], \Bbb Z[\frac{1+\sqrt {-7}}{2}], \Bbb Z[\frac{1+\sqrt {-11}}{2}]$$
have norms $a^2 + ab + b^2,a^2 + ab + 2 b^2,a^2 + ab + 3 b^2$.
and it seems primality or divisors of them is perhaps not completely unrelated or independant. ( for instance if one of them is a square, then the previous one is that square minus $b^2$ what implies probably composite since $(c^2 - b^2) = (c+b)(c-b)$ )
This post asks somewhat a multitude of related questions, selected pairs of rings and visuals, so multiple answers are possible.
To avoid not having a main question I post this :
MAIN QUESTION :
What is known about the equation :
$$ p_1 = a^2 + ab + b^2, p_2 = a^2 + ab + 2 b^2, p_3 = a^2 + ab + 3 b^2 $$
where $p_1,p_2,p_3$ are odd primes and $a,b$ are positive integers ?
or equivalently the "simultaniously primes " of $\Bbb Z[\frac{1+\sqrt {-3}}{2}], \Bbb Z[\frac{1+\sqrt {-7}}{2}], \Bbb Z[\frac{1+\sqrt {-11}}{2}]$ ?
And how does it look like in the cartesian plane with $a,b$ on the 2 axises ? Do we get an interesting pattern ? Some conjectures ? Does it hold if we zoom out ?
As Gerry Meyerson correctly points out $b$ needs to be a multiple of $3$ so we actually get the diophantine equations (setting $b = 3B$) :
$$ p_1 = a^2 + 3aB + 9 B^2, p_2 = a^2 + 3aB + 18 B^2, p_3 = a^2 + 3aB + 27 B^2 $$
and ofcourse $a,b$ or $a,B$ need to be relatively prime.
RELATED SECONDARY QUESTION :
Is the number of such simul primes within radius $x$ very close to this estimate or is it a bit off ? :
$$ \pi_3(x) = \frac{x}{(\ln(a^2 + ab + b^2)-1)(\ln(a^2 + ab + 2 b^2)-1)(\ln(a^2 + ab + 3 b^3)-1)}$$
(this is similar to prime constellation estimates, but in this case unlike prime twins we know they exist and have good idea about their density )
how good is this estimate ?
