1
vote
0answers
48 views

Finding the number of elements in $\left(ℤ[i]\right)_m$

If $m$ = $m_1$ + $m_1i$ ∈ $ℤ[i]$, is there a general formula to find the number of elements in $\left(ℤ[i]\right)_m$?
0
votes
1answer
39 views

Number theory proof regarding norms

How would you prove that if $x$ is a prime in $ℤ[i] \Longleftrightarrow$ $N(x)$ is a prime in $ℤ$ N(x) represents the norm of x.
5
votes
1answer
178 views

Why all composite numbers have this property?

Define $f(n)=\sum\limits_{A \in S} f_{1}(n,A),\ n>2,\ n \in \mathbb{Z}$, where $S$ is the power set of $\{\frac{1}{2},\cdots ,\frac{1}{n-1}\}$. Define $\ f_1(n,\varnothing)=1,\ ...
0
votes
2answers
83 views

Closure of Integers under multiplication and rational exponentiation

What is the closure of the Integers under a finite number of multiplications and rational exponentiations? For example, $3^{1/2}$, $i = -1^{1/2}$, and $\frac{-1+i \sqrt(3)}{2} = 1^{1/3}$ all in this ...
0
votes
1answer
59 views

A complex series with exponentials

I have tried to solve this type of series : $$\sum \frac{e^{i\, u(n)}}{v(n)} $$ For some $u,v$ an Abel Transform allow to find convergence, but for $u(n)=n^2$ and $v(n)=n$ I can't find an argument. ...
2
votes
1answer
95 views

How to find all pairs $(a, b)$ s.t. $(a^2+b^2)/\gcd(a,b) \leq n$ for constant $n$?

Any help is appreciated, this is for my work on http://projecteuler.net/problem=153. Also posted here
1
vote
2answers
91 views

Results Analogous to the Two and Four Square Theorems.

A result that arises out of the study of $\mathbb{Z}[i]$ is that the following are equivalent for integer primes p: 1) $p\equiv 1$ (mod 4) or $p=2$ 2) $\exists a,b\in\mathbb{Z}$ such that ...
44
votes
4answers
573 views

Integers $n$ such that $i(i+1)(i+2) \cdots (i+n)$ is real or pure imaginary

A couple of days ago I happened to come across [1], where the curious fact that $i(i-1)(i-2)(i-3)=-10$ appears ($i$ is the imaginary unit). This led me to the following question: Problem 1: Is $3$ ...
4
votes
1answer
108 views

Are there any arguments against the Riemann hypothesis?

We all know the well known Riemann hypothesis that the zeroes of the Riemann-zeta function have real part $1/2$ seems to hold (as far as I know) for all prime numbers. I was curious if there were any ...
1
vote
0answers
58 views

Zeta zeros by recurrence of zeta function, but this is useless isn't it?

One more useless question of mine can't do this site any harm. So here we go. The following Mathematica program converges to most of the riemann zeta zeros, by using an approximation as a starting ...
0
votes
0answers
136 views

Counterexample for conjugate rules in $\mathbb{Z}[\sqrt[4]{2}]$

We all know that, in a field, such as $\mathbb{Z}[i]$ or $\mathbb{Z}[\sqrt[4]{2}]$, conjugate properties that as $ \overline {u \cdot v} = \overline {u} \cdot \overline {v} $ hold. Furthermore, ...
0
votes
2answers
63 views

When $a$ is even, the difference between $(a/2) \mod N$ and $(a \mod N)/2$?

folks. Could I ask for your help? Let $N$ be a positive integer and $a$ be an even integer, i.e., $a=2x$ for an integer $x$. Then think of $W_N^{\frac{a}{2}}$, where $W_N=e^{j\frac{2\pi}{N}}$. ...
4
votes
2answers
195 views

How to evaluate a zero of the Riemann zeta function?

Here is a super naive question from a physicist: Given the zeros of the Riemann zeta function, $\zeta(s) = \sum_{n=1}^\infty n^{-s}$, how do I actually evaluate them? On this web page I ...
5
votes
1answer
158 views

Do there exist complex algebraic $α,β$ such that $α^β=π$ or $α^β=e$?

Given the algebraic operations and complex exponentiation $(a+bi)^{c+di}$ and logarithm, is it possible to derive $\pi$ and $e$? If one is derivable then so should be the other, as $e^\pi = ...
4
votes
3answers
158 views

Why don't we define division by zero as an arbritrary constant such as $j$? [duplicate]

Why don't we define $\frac 10$ as $j$ , $\frac 20$ as $2j$ , and so on? I know that by following the rules of math this eventually leads to $1=2$ , but we could make an exception and say that $j$ is ...
0
votes
1answer
90 views

Solving complex linear congruences

Find $x \in \mathbb{Z}[i]$ such that: $(1+2i)x \equiv 1 \mod 3+3i$ How would you go about doing this? Best I can think of is keep guessing....
1
vote
1answer
187 views

extended Euclidean (xgcd) in quadratic integer rings

Given a discriminant $D < 0$, I have the quadratic imaginary field $\mathbb{K} := \mathbb{Q}(\sqrt{D})$. And the quadratic integer ring is given by $\mathcal{O} = \mathbb{Z} + \mathbb{Z} \frac{D + ...
5
votes
0answers
291 views

Natural extension of the divisor function $\sigma$ to the complex integers $\mathbb{Z}[i]$

The divisor function $\sigma$ of a positive integer $n$ is defined as the sum of the positive divisors of $n$. It turns out that this definition is equivalent to ...
5
votes
1answer
108 views

A case where $z^z = 0$ where $z$ is complex number

Is there any case where $z^z = 0$ where $z$ is complex number? The case excludes the case where $z=0$.
1
vote
1answer
88 views

3 complex-variable equation

Moderator Note: This is a current contest question on Brilliant.org. $x,y,z$ are complex numbers satisfying $$ \begin{align} x+y+z & =1\\ x^2+y^2+z^2 & =2\\ x^3+y^3+z^3 & =3 ...
2
votes
0answers
86 views

convergence of a sum with zeroes of zeta function

Can it be proved that the sum of this series is smaller than $x$? $$ \sum_{\zeta(a+ib)=0}u_{a,b}(x)\lt x, $$ for all $x$, with $$ ...
0
votes
2answers
49 views

Proving two Complexes' Numbers Properties

I'm having problem working with complex number on this question and was wondering if someone can walk through with me their reasoning on how to solve this/these types of questions. Thanks in advance! ...
3
votes
3answers
137 views

How to find $(-64\mathrm{i}) ^{1/3}$?

How to find $$(-64\mathrm{i})^{\frac{1}{3}}$$ This is a complex variables question. I need help by show step by step. Thanks a lot.
3
votes
1answer
213 views

Linear relations satisfied by nth root of unity

How do you characterize all the linear relations satisfied by $n$th roots of unity with real, integral and non-negative integral coefficients? Here are two examples for 3rd and 4th root: Let ...
1
vote
2answers
67 views

If $N(\alpha)$ = $p$ where $p$ is an odd prime, then is ($\alpha, \bar\alpha$) = $\mathbb{G}$?

Here, $N(\alpha)$ stands for the norm of $\alpha\in\mathbb{G}$, $\mathbb{G}$ is the set of Gaussian Integers, and ($\alpha, \bar\alpha$) is the ideal generated by $\alpha$ and $\bar\alpha$. In other ...
1
vote
1answer
97 views

Can a convergent sum using only integers produce a complex result?

We use this function to define the boundaries for the product in the denominator: $$f(\text{n$\_$})\text{:=}\frac{1}{8} \left(2 n (n+2)-(-1)^n+1\right)$$ We calculate the infinite sum: $$\sum ...
2
votes
1answer
158 views

Next generation numbers

1: Discovering of negative numbers. Assume a and b are positive integers $x+a=b$ ----> if $b>a$ then $x$ is positive integer $x+a=b$ ----> if $b=a$ then $x=0$ $x+a=b$ ----> if ...
2
votes
1answer
545 views

primitive roots of unity

How to prove that if $\theta _1,\theta _2,\theta _3$ be the arguments of the primitive roots of unity, $\sum \cos p\theta = 0$ when $p$ is a positive integer less than $\dfrac {n} {abc\ldots k}$, ...
10
votes
2answers
476 views

Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$

1) Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$. Prove: Is it true that $a \in \mathbb{Q}$? 2) Suppose $a \in \mathbb{C}$, ...
0
votes
1answer
84 views

Representing complex numbers with nested exponentiation of rationals

Define $L_0=Q$ $L_1=\lbrace x \in C; e^{x} \in L_0 \rbrace$ $L_{-1}=\lbrace x \in C; \ln{x} \in L_0 \rbrace$ $L_{n+1}=\lbrace x \in C; e^{x} \in L_n \rbrace$ $0$ is in $L_1$ and $L_0$. Do any ...
1
vote
1answer
121 views

Do these zeros have real part equal to $0$?

I guess this is a known result but I could not find it on the Internet. Consider these equations formed from the reciprocals of the divisors of $n$ raised to a complex number $s=a+ib$ : ...
20
votes
2answers
823 views

Complex solutions for Fermat-Catalan conjecture

The Fermat-Catalan conjecture is that $a^m + b^n = c^k$ has only a finite number of solutions when $a, b, c$ are positive coprime integers, and $m,n,k$ are positive integers satisfying $\frac{1}{m} + ...
5
votes
2answers
188 views

Lerch-$\small \zeta(\varphi,0,-n)$ of integer *n* purely real and imaginary ($\small \zeta_\varphi (-n)^2 $ is real) for $\small n \ge 2$?

Are the Lerch-$\zeta(\varphi,0,-n) $ of integer n (for shortness I use the notation of my earlier question $\small \zeta_\varphi(-n)$) periodically purely real and imaginary: $\zeta_\varphi (-n)^2 $ ...