# Questions tagged [eisenstein-integers]

For questions about Eisenstein integers. The Eisenstein integers are the complex numbers of the form $a+b\omega ,$ where $\omega=e^{2\pi i/3}$ is a primitive $3$rd root of unity. Sometimes they're also called the Eulerian integers. They form a subring of $\Bbb Q(\omega).$

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### Is $c = \frac{\sqrt 3}{4} \frac{\pi}{4} \prod_{p = 2 \mod 3} \sqrt{\frac{p^2}{p^2-1}} \prod_{q = u^2 + 3 v^2} \sqrt{\frac{q^2}{q^2-1}}$?

Consider the sum of $2$ squares and Gauss circle problem https://en.wikipedia.org/wiki/Gauss_circle_problem and also The Landau-Ramanujan Constant that relates to the sum of 2 squares. See : http://en....
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### Is every sufficiently large gaussian integer the sum of 3 cubes ? $a + b i = (c + di)^3 + (e + fi)^3 + (g + hi)^3$?

Is every sufficiently large gaussian integer the sum of $3$ gaussian cubes ? In other words, $$a + b i = (c + di)^3 + (e + fi)^3 + (g + hi)^3$$ for given integers $a,b$ with $a^2 + b^2 > Q$ can ...
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### Law of cubic reciprocity for a primary prime of norm 3

I am currently studying "A Classical Introduction to Modern Number Theory", a book by Kenneth Ireland and Michael Rosen. In that, we have the following two theorems: $(1)$ The Law of Cubic ...
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### Cubic non-residue calculation

I am currently studying Cubic residue characters from Kenneth Ireland and Michael Rosen's "A Classical Introduction to Modern Number Theory", and this is the definition given in the book: If ...
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### Eisenstein integers with norm value of 5

I am trying to find out the Eisenstein integers that have norm values of 3,5,7.... I want to see if there is some pattern. For the norm value of 3, I was able to find six Eisenstein integers. For the ...
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### Let $\pi \in \mathbb{Z}[\omega]$ be a prime. Then $\frac{\mathbb{Z}[\omega]}{\pi \mathbb{Z}[\omega]}$ has $N(\pi)$ elements

I am currently reading cubic and biquadratic reciprocity from Kenneth Ireland and Michael Rosen's, "A Classical Introduction to Modern Number Theory" have a doubt in proposition 9.2.1. We ...
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### For Eisenstein numbers $a + b\frac{1+\sqrt{3} i}{2}$ with $0 \le a, b \le n$ how can I efficiently identify all of the primes?

This answer to determining if a coincident point in a pair of rotated hexagonal lattices is closest to the origin? teaches that my real question was how to find if two Eisenstein integers are coprime. ...
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### If $x \equiv 2 \pmod 3$, show that $x=\pm x_1 x_2 \cdots x_t$, where $x_i\equiv 2 \pmod 3$

This question comes from A Classical Introduction to Modern NT (Ireland & Rosen), chapter $9$ $\#17$. Note that $D$ is the set of Eisenstein integers. Exercise 9.17 An element $x \in D$ is called ...
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### Are all points on the shore of Gosper Island either transcendental or Eisenstein rational?

I'm working on a program which calculates in the flowsnake base (2.5-√-0.75, with cyclotomic digits) and I've come up with some observations and questions about this base representation. The program ...
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### On primality of an element in the ring $\mathbb Z[\omega]$ [duplicate]

The problem is: Show that if $p(\neq 3)$ is a prime in $\mathbb Z$ and $p \neq a^2+b^2-ab$ for any $a,b \in \mathbb Z$, then $p$ is a prime element in the ring $R = \mathbb Z[\omega]$. My approach: I ...
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### using eisenstein integers to solve diophantine equations

Earlier, I asked a question; this is a question regarding an answer I received to it. Apparently when $u^2 - 12p^2 = -3$, with $p$ a three digit prime, there is some sort of recursion of the values of ...
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### $x^3 + y^3 = p^2$ over the integers

$x^3 + y^3 = p^2$ has a solution over the integers for some three digit prime p. Find all p that satisfy. The first thing I did was factorize the left hand side, getting $(x+y)(x^2 - xy + y^2) = p^2$ ...
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### How to prove that $\mathbb{Z}[\exp(\frac{2\pi i}{3})]$ is a Euclidean domain? [duplicate]

To give the context I am currently studying Gaussian Integers, and I have of course studied rings. The full question is: Given $\rho = \exp(\frac{2\pi i}{3})$, show that the ring $R = \mathbb{Z}[\rho]$...
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### Conductor of $\mathbb Q(\omega,\sqrt[3]{\pi})/\mathbb Q(\omega)$ for nonprimary $\pi$

I have recently been playing around with abelian extensions and I have have found myself playing with Magma and computing conductors of cubic extensions of $F=\mathbb Q(\omega)$, where $\omega$ is a ...
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