# Questions tagged [nt.number-theory]

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

2,960 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
180 views

### Invariance of the (Liouville-Roth) irrationality measure under rational Möbius transformations

For a real number $x$, we define the (Liouville-Roth) irrationality measure of $x$, here denoted by $\mu(x)$, as the infimum, with respect to the poset $(\mathbb{R}_0^+ \cup \{\infty\}, \le)$, of the (...
287 views

### On unique solutions to linear diophantine equations

Let $\sum_{i=1}^k a_ix_i = N$ be the equation with $a_i \in [2^t,2^{t+1}]$ being distinct primes. If we seek unique solutions $x_i\in R_i = (0,a_i)\cap \Bbb Z$, then in general it is not possible. ...
98 views

159 views

### Uniqueness of decomposition of completely reducible representations

Let $X$ be a smooth, separated scheme of finite type over $\mathbb{F}_q$ where $q=p^r$ for some $r>0$. Let $gcd(l,p)=1, \rho:W(X) \to GL_r(\mathbb{Q}_l)$ be a Weil representation which is semi-...
87 views

### Existence of primitive totally real number fields

For any positive integer n, is there always a primitive totally real number field of degree n over $\mathbb Q$? I am new to this field. Please forgive me for asking such an elementary question.
139 views

### Simple diophantic equation

We are looking for all integer solutions for the equation $a^b+1=b^a$. We conjecture that there are only the solutions $(0,b),(1,2),(2,3)$. It is easy to see, that if a is odd and b even, there is ...
97 views

### Equation system on Z[x]

Can we solve this equation system when $n=2,3,4,\cdots \$ is given? $$F(x)+F((2^n-1)x) = (G(x))^nx^g$$ $$2F(2^{n-1}x) = (H(x))^nx^h$$ While $F,G,H \in \mathbb{Z}[x]. g,h \in \mathbb{N}.$
180 views

### A Number Theory Question about Polynomial

Now we have a Polynomial $P(n)$ on $\mathbb{Z}[x]$. It can't be wriiten as $P(n)=F(n)G(n)$ while $F(n),G(n) \neq 1$. Is it right that for any $P(n)$, there is a $n$ such that $P(n)$ is a prime? Is it ...
324 views

### Does there exist a Weierstrass/Hadamard factorization for $\chi(s)-1$?

Would like to build once more on this question. Take $s=\sigma + ti, s \in \mathbb{C}, 0<\Re(\sigma)<1$. Let's assume it is proven that: $$\zeta(1-s) - \zeta(s)$$ has all its zeros on the ...
277 views

304 views

### Rational Integer Solutions of a Linear Diophantine Equation of Cyclotomic Integers

Hi everyone. I am working with lattices in $\mathbb{C}$, and I want to know whether a certain vector is an element of the lattice. In particular, suppose my lattice vectors are $a$ and $b$ and I ...
331 views

### What is the mean-value of a particular exponential sum related to the non-trivial zeros of Riemann's zeta function?

This question arose from an earlier one and the MO user's useful answers there: What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (which is not a ...
184 views

### Number of biquadrates mod n

Is there an explicit formula for the number of fourth powers mod n? Finch & Sebah  give theorems, partially folklore, for squares and cubes mod n, but I don't know of a similar formula for ...
259 views

### Computing the function field of a curve given as a subvariety of the Jacobian of its cover or merely the degree of the covering

I read following paragraph from: G. Tamme, Teilkörper höheren Geschlechts eines algebraischen Funktionenkörpers, Arch. Math. 23 (1972), 257--259 Here $C$ is a curve of genus $\ge 2$ and $J$ is the ...
275 views

### Exponential Diophantine $\prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$ ,$e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$

Does the exponential diophantine equation $$\prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$$ with $x_i, y_i,z_i$ coprime and $e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$ have solutions?...
383 views

### Boxing the Rational Box

You surely all know the Perfect Cuboid problem. Here is a bit of pondering on the Euler box (space diagonal doesn't have to be rational). We start with the generators p,q,r and the surface (p^2-1)(q^...
150 views

### Number of elements of the same norm in a quadratic ring

Consider the ring $Z(\sqrt D)$ where $D$ is a negative integer. Let $N$ be a positive integer. I wonder if there is a formula/estimate for the number of elements of norm exactly $N$ ?
267 views

### On a variant of the equation $\sigma(x)=2^n$

It is a nice exercise to prove that the only solutions (positive integers $x$) of the equation on the title are products of Mersenne primes; with all exponents equal to $1$. ((see also: A046528 in ...
254 views

### Solving in positive integers an equation containing exponentials

I posted this question on Math.SE and it has been a while that no non-trivial hint has been suggested. I hope MOers will have something to say or will solve it entirely. Here is the question copied ...
411 views

### Sums of inverses of odd'' elements modulo $p$

In a letter to Dirichlet, Gottold Eisenstein stated the congruence: $$q(u) \equiv u - \frac{u^2}{2}+ \frac{u^3}{3} - \frac{u^4}{4} + \cdots + \frac{u^{p-2}}{p-2}-\frac{u^{p-1}}{p-1} \pmod{p}$$ ...
308 views

### Estimating a multinomial sum

I have the following sum \begin{equation} \sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda} \...