Questions tagged [gaussian-integers]
This tag is for questions relating to the Gaussian integer, which is a complex number $~z=~a~+i~b~$ whose real part $~a~$ and imaginary part $~b~$ are both integers.
355
questions
6
votes
1
answer
66
views
Can we (almost always) walk from one Gaussian non-prime to another?
This is a plot of the Gaussian primes. They get sparser as you move further from $0$, so it looks like if you start on one of the white squares you could travel to any other white square (almost) ...
- 5,633
3
votes
2
answers
111
views
Homomorphism between $\mathbb{Z}[i]$ and $\mathbb{ Z/(p)}$
I'm trying to construct a non-zero ring homomorphism between $\mathbb{Z}[i]$ and $\mathbb{ Z/(p)}$. The question is for what $p$ is it possible. I managed with one direction:
$f(1) = f(-i^2)=-(f(i))^2 ...
- 95
0
votes
1
answer
50
views
An estimation of Bezout Coefficients(produced by Extended Euclidean Algorithm) on Gaussian integers
Problem: Suppose that for two given Gaussian integers $a$ and $b\ (|a|>|b|>0)$, there exists $a_0$ such that the remainder of the Euclidean division of $a_0$ by $a$ is exactly $b$. If it takes $...
- 49
-3
votes
0
answers
46
views
A property of the steps of Extended Euclidean Algorithm on Gaussian integers [duplicate]
Suppose that we have a computer program, if we give it two Gaussian integers $a, b (|a|>|b|>0)$, it will return three numbers $x,y,n$, where $x$ and $y$ are the Bezout coefficients produced by ...
- 49
0
votes
0
answers
87
views
How is the following Matsubara sum solved? $\sum_{n} \frac{2x}{2i(2b - i\omega_{n})^{2} + x^2}$
$\sum_{n} \frac{2x}{2i(2b - i\omega_{n})^{2} + x^2}$, where $b$ represents some chemical potential and $\omega_{n}$ are the bosonic frequencies. The conflict for me is the factor $2i$ that appears in ...
- 11
1
vote
0
answers
37
views
Question about prime elements in the gaussian numbers
Let $\mathbb{Z}[i]:=\{a+ib : a,b \in \mathbb{Z}\}$ denote the gaussian numbers. Let $N$ denote the euclidean function $N(z):=z \overline{z}$ and $\mathbb{P}$ denote the positive prime numbers.
Reading ...
- 609
2
votes
1
answer
92
views
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 ...
- 16.4k
0
votes
0
answers
19
views
How do I use the gaussian divisors formula?
For an integer z,
$$
z = \epsilon \prod_i p_i^{k_i},
$$
where $\epsilon$ is and a unit and every $p_i$ is a Gaussian prime in the first quadrant then the sum of the Gaussian divisors is
$$
\sigma_1 (z)...
-3
votes
1
answer
67
views
Real numbers extend to complex numbers. Why do (real) integers extend to 'Gaussian integers' instead of 'complex integers'? [closed]
Real numbers extend to complex numbers are $\mathbb C := \{ a+bi;a,b \in \mathbb R \}$. So why isn't the set $\{a+bi;a,b \in \mathbb Z \}$ called complex integers instead of Gaussian integers, an ...
- 13.7k
0
votes
2
answers
65
views
prime factorization in $\mathbb{Z}[i]$ [duplicate]
We were asked to show where the following reasoning goes wrong. Since $1+i$ and $1-i$ are prime elements in $\mathbb{Z}[i]$, the equation $$(-i)(1+i)^2=(1+i)(1-i)=2$$ show that unique prime ...
- 103
0
votes
0
answers
39
views
Cauchy principal value for a Gaussian integral of a rational function
I wish to calculate integrals of the following form, all Gaussian integrals of a rational function, on the entire real domain:
$$ I_{k,n} = \int_{-\infty}^{\infty} dx \frac{e^{-x^2/2}}{\sqrt{2\pi}} \...
- 395
0
votes
0
answers
8
views
Zariski-density on almost diagonal embedding
It is not hard to see that the Gaussian integers $\mathbb{Z}[i]$ are Zariski-dense inside $\mathbb{C}$, seen as an affine space over $\mathbb{C}$. Consider now the set $$D = \{(z,\overline{z}) \in \...
- 2,965
9
votes
2
answers
170
views
Remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\Bbb{Z}[i]$
Find the possible remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\mathbb{Z}[i]$.
I'm having a hard time understanding remainders in $\mathbb{Z}[i]$. I'm gonna write my solution to the ...
- 464
2
votes
1
answer
87
views
Question about the prime elements of $\mathbb{Z}[i]$.
I am learning about gaussian integers and I have a few questions about the following argumentation.
What are the prime elements in $\mathbb{Z}[i]$? We remember that only
the units $+1,-1,+i,-i$ in $\...
- 609
0
votes
1
answer
39
views
Ring structure of $\mathbb{Z}/(n^p+1)\mathbb{Z}$
I encountered the ring $\mathbb{Z}/(n^p+1)\mathbb{Z}$, where $n$ is some positive integer and $p\in\{2,3,5,...\}$ a prime and I am wondering whether there is a difference in the structure when $p=2$ ...
- 1,271