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

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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 (...
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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. ...
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98 views

Sequences sharing some primitive prime divisors

Let $q=p^\alpha$ and $q'=p'^\alpha$. Moreover, define $r_i$ and $u_i$ as primitive prime divisor of $q^i-1$ and $q'^i-1$, respectively. Let $\{r_1\}=\{u_1\}$, $\{r_2\}=\{u_2\}$, $\{r_3\}=\{u_3\}$, $\{...
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146 views

Derivative of a function related to Dedekind zeta function

Lef $K$ be an algebraic number field of degree $[K:\mathbb{Q}]=n$. For simplicity suppose $K$ is totally real. Define $f(s) = \zeta_K(s) \zeta(1-s)^{n-1}$ where $\zeta = \zeta_{\mathbb{Q}}$. From the ...
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318 views

Motivation behind the appearance of Bessel functions in partial trace formulas

Bessel functions occur naturally on the Kloosterman side (or geometric side) of Petersson's formula and Kuznetsov's formula. Is there an intuitive explanation for their appearance? For instance, is ...
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106 views

Diophantine question (again related to knot theory)

I finally managed to compute the dimensions of some higher irreps of my super-duper-generalized $E_7$ series. Given $Dim(V)=i,Dim(J)=j$ (where $V$ and $J$ are defining and adjoint irrep, and $i,j$ are ...
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145 views

Is there a reference book for the duality between the genus of function fields and the discriminant of number fields?

Bjorn Poonen mentions in his "Lectures on rational points on curves" the analogy between the genus of a function field and the discriminant of number fields. I'm looking for a reference book for this ...
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115 views

field of definition of abelian varieties with extra endormorphism

Let $A$ be a complex abelian variety such that $\mathrm{End}(A)$ is strictly bigger than $\mathbb{Z}$. Question: Is is true that $A$ is defined over $\overline{\mathbb{Q}}$? This is of course what ...
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167 views

Prime-Counting Function

Would a summation of floor[cos^2π(((j-1)!+1)/j)] from j=1 to x be the same as π(x)?
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304 views

Solution of a special class of Diophantine Equations

Suppose A is Diophantine Equations which have a variable x,such that every other variable has function of x the variable as it's solution.For example,a Diophantine equation in three variables has ...
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271 views

r-torsion points on elliptic curve on finite field

Let $E(\mathbb{F}_q)$ - elliptic curve, $r$ is prime, $|E(\mathbb{F}_q)[r]| > 1$. Let $r | q-1$. Is it true that $|E(\mathbb{F}_q)[r]| = r^2$?
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334 views

Analytic rank of an elliptic curve with algebraic rank 0

Let $E$ be an elliptic curve over $\mathbb{Q}$ with algebraic rank 0. Is there any way, one can argue that the analytic rank must be even? Of course this would follow from the standard conjectures, ...
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397 views

Solutions to a quadratic congruence

Fix an odd prime $p$. Let $\alpha = (\alpha_0,\dots,\alpha_k)$ be a solution to the congruence $\sum_{i=0}^{k} \alpha_i^2 \equiv x \mod p$. Now consider the number $N_\alpha$ of solutions to the ...
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95 views

Asymptotic inverses of asymptotic functions

The prime number theorem states that two functions are asymptotic. Their inverses (as functions of an integral variable) are also asymptotic. In general, under what conditions are the inverses of ...
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130 views

Find polynomial in finite field

We have $A$, $B \in GF(q^k)$ We want to find polynomial $h \in GF(q)[x]$ where $h(A) = B$ What is the lowest degree of $h$? How to find $h$ with the lowest degree and what is complexity of this ...
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305 views

Definitions for Oddness

In the thread Even Xor Odd Infinities I defined odd models of Modular Arithmetic (MA) as models satisfying the axioms of MA and two first order statements. Even XOR Odd Infinities? $O1) \forall x(x=0 ...
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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-...
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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.
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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 ...
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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 $F,G,H \in \mathbb{Z}[x]. g,h \in \mathbb{N}.$
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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 ...
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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 ...
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277 views

zeros of a polynomial in a finite field

Hi everyone. While looking for examples of asymptotically bad towers of function fields over a finite field $F_q$ defined by a Kummer equation of the form $y^m=f(x)$ with $p\equiv 1\mod m$ where $p=...
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149 views

Weaker conditions for potential good reduction of Abelian varieties

We are concerned with slight weakening of a result of the Serre-Tate paper "Good reduction of abelian varieties" google. I think the title of the question conveys what we are in here for. So, I'll ...
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235 views

Ring of Lipschitz Integers

Let $L$ be the ring of Lipschitz Integers and $a, b, c, d\in L$. Considere $L$ as a left $L$-module and let $(a), (b), (c), (d)$ be the left submodules generated by $a, b, c,$ and $d$, respectively. ...
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121 views

Sets composed of arithmetic progressions

Take a generalized arithmetic progression (GAP) $$P_{n} = \lbrace a + x_{1}r_{1} + x_{2}r_{2} + \ldots + x_{n}r_{n} | 0\leq x_{i} < k_{i} \rbrace = \lbrace a + \sum_{i=1}^{n} x_{i}r_{i} | 0\leq x_{...
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125 views

Lower Bounds on Binary Forms of degree m

The following paper http://www.math.leidenuniv.nl/~evertse/05-discres.pdf looks at the binary form $\sum \alpha_ix^iy^{m-i} where i ranges from 0 to m. It then defines the discriminant. It's well ...
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201 views

A question on Selberg's sieve

If we define $$D=D(z,w)=\sum_{d\vert P_z, d\leq w}\frac{1}{\phi(d)}$$ $$\lambda(k)=\frac{k}{D} \sum_{k\vert d,d\vert P_z , d\leq w}\frac{\mu(\frac{d}{k})\mu(d)}{\phi(d)}$$ $$p(d)=\sum_{lcm(d_1,d_2)=d ...
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432 views

Upper bound for generalized harmonic number wih negative exponent

Hi, I need an upper bound for the generalized harmonic number with negative exponents, i.e: $$H_{n,r}=\sum_{k=1}^n \frac{1}{k^r}$$ where $r<0$ especially, I need a bound for $$r=-\frac{1}{2}$$ ...
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597 views

Igusa model of modular curves

I would like to know what the "Igusa model" of the modular curve is and basic properties about it. Can someone point me to a reference?
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124 views

A kind of orthogonal subgroup

Let $n$ a positive integer and $k \in \mathbb{Z}^n$ such that for all integer $a \geq 2$ and $h \in \mathbb{Z}^n$ we have $k \neq ah$. Here $\cdot$ is the scalar product. Is it true that $\{x \in \...
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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 ...
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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 ...
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184 views

Number of biquadrates mod n

Is there an explicit formula for the number of fourth powers mod n? Finch & Sebah [1] give theorems, partially folklore, for squares and cubes mod n, but I don't know of a similar formula for ...
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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 ...
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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?...
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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^...
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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 $N$ ?
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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 ...
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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 ...
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411 views

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

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

Looking for product of symmetric polynomials evaluated at roots of unity

Consider $a_{1} = \alpha^{N-n}, a_{2} = \alpha^{N-n+1}, a_{3} = \alpha^{N-n+2}, a_{4} = \alpha^{N-n+3}, \cdots, a_{n} = \alpha^{N-1}$ where $\alpha$ is a complex $N$th root of unity where $N = 2 + (n-...
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303 views

Are the largely composite numbers the same as the fully composite numbers?

Let $d(n)$ be the number of divisors of $n$. Let $p(n)$ be the product of the divisors of $n$. Ramanujan called a number $n$ largely composite if $d(n) \ge d(m)$ for $m < n$. Let's call $n$ fully ...
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294 views

L-series method, how far can it go?

Using some suitable L-series for some appropriate ray class group, one can find the Dirichlet density of some set of primes. One can conclude that this set of prime is infinite as long as the density ...
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550 views

Reducing two variable linear Diophantine equation to modular inversion

I'm in the field of secure multiparty computation using Homomrphic encryption or secret sharing. I want to implement a secure protocol to compute the GCD of two encrypted numbers. To calculate the ...
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311 views

When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, a square?

It is easy to show that the following problems are equivalent. a. When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, and $n$ any integer, a square? and b. When is $X^2-PY^2=k$ ...
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239 views

question regarding double summations

I'm looking for a reference and/or table for double summations. The sum I'm trying to compute is $$\sum_{k=1}^\infty \sum_{m=1}^\infty \frac{1}{km(ak^2+bm^2)}$$ for real numbers $a$, $b$.
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169 views

Measuring for how many $p$ a number is a generator for $(\mathbb{Z}/p\mathbb{Z})^*$

If $p$ is a prime number, we denote by $(\mathbb{Z}/p\mathbb{Z})^*$ the multiplicative group of the field $\mathbb{Z}/p\mathbb{Z}$. It is known that $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic. For $A\...
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44 views

Ramanujan primes in short intervals

I'm curious to know if is in the literature a similar/analogous statement about Ramanujan primes (this Wikipedia or [1]) in short intervals than those that refers the Wikipedia Bertrand's postulate ...