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
15,955
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Finite generation for a restricted ramification idele module
Let $k$ be a number field, let $\bar k \subseteq \mathbb{C}$ be a fixed algebraic closure of $k$, and let $S$ be the set of infinite primes of $k$. Denote by $k_S$ the maximal extension of $k$ inside $...
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Is it expected that the mod $p$ representation determines a normalized Hecke newform of fixed weight for p large enough?
Mazur's conjecture on the image of Galois representations of Elliptic curves states that for $N$ large enough there is a unique elliptic curve $E$ over $\mathbb{Q}$ giving rise to a fixed mod $N$ ...
user130124
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1
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Asymptotic behavior of sum linked with Lagrange interpolation
I already asked this a few weeks ago with no answer, so let me formulate differently.
In performing Lagrange interpolation with nodes 1/n, one encounters the sum
$$S(f)=\dfrac{1}{N!}\sum_{n=0}^N(-1)^{...
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1
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Does the Riemann Xi function possess the universality property?
Here is the question.
Does the Riemann Xi function possess the universality property, or something similar to Voronin's universality property?
Here is why the answer to this question is important. ...
4
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0
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holomorphic continuation of motivic $L$-functions
The question is rather easy to formulate: when is the $L$-function of a pure motive over $\mathbb{Q}$ expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex ...
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3
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2
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460
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Is it possible to find the maximum value of a sum of absolute differences?
Let $a_1$, $a_2$, …, $a_n$ and $b_1$, $b_2$, …, $b_n$ be $2n$ strictly positive integers not greater than $M$, with $M$ a given positive integer, such that $$a_1+ a_2+ \dotsb+ a_n=b_1+ b_2+ \dotsb+ ...
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3
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Lagrange four squares theorem
Lagrange's four square theorem states that every non-negative integer is a sum of squares of four non-negative integers. Suppose $X$ is a subset of non-negative integers with the same property, that ...
- 1,651
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1
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200
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The largest number $y$ such that $(x!)^{x+y}|(x^2)!$
Since the multiplication of $n$ consecutive integers is divided by $n!$, then $(n!)^n|(n^2)!$ with $n$ is a positive integer.
Are there any formula of the function $y=f(x)$ that shows the largest ...
- 501
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1
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$\sum_{n=0}^{N-1}\exp(-i\frac{\pi}{N}n^2)=\sqrt{N}\exp(-i\frac{\pi}{4})$
The question is: N is an even positive integer, then $\sum_{n=0}^{N-1}\exp(-i\frac{\pi}{N}n^2)=\sqrt{N}\exp(-i\frac{\pi}{4})$. I thought the terms on the left are the solutions set of some polynomial ...
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Are there infinitely many integers $m$ such that $a+m$ divides $a^m+1$?
Let $a$ be a positive integer.
Does there exist a positive integer $m$ other than 1 such that: $$a+m \ |\ a^m+1 \ $$ If not, what are the conditions of $a$ for $m$ to exist?
If there exist ...
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Does the equation $(xy+1)(xy+x+2)=n^2$ have a positive integer solution?
Does there exist a positive integral solution $(x, y, n)$ to $(xy+1)(xy+x+2)=n^2$? If there doesn't, how does one prove that?
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Rational points on a weighted projective surface
The equation
$$\displaystyle y^2 = f(x_1, x_2, x_3)$$
with $f$ a non-singular quartic form in three variables defines a del Pezzo surface of degree 2. I am interested in the similar construction
$$\...
- 25.5k
4
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Congruence for the product of quadratic residues + the product of quadratic non-residues
My question has been here on MSE for a long time, but it has not received a full answer. I bring it here:
Find a prime $p$ such that $p \equiv 1 \bmod 4$ and such that the
product in the range $[...
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Two reasons why the Collatz conjecture could fail
Let $\mathbb{N}$ denote the set of positive integers. The Collatz function $f:\mathbb{N}\to\mathbb{N}$ is given by $f(n) = n/2$ for $n$ even and $f(n) = 3n+1$ for $n$ odd. Given $k\in\mathbb{N}$ we ...
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Local factors determine Weil representations - proof of the Artin representation case
This post can be seen as a continuation of this post I created on MathOverflow.
I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...
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Local properties of Galois representations attached to torsion classes
$\DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\GL}{GL} \newcommand{\F}{\mathbb{F}} \newcommand{\p}{\mathfrak{p}} \DeclareMathOperator{\Sym}{Sym}$
Let $F$ be a number field, and let $\Gamma$ be ...
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A question about distribution of fractional part of $2^k\alpha$
Let $\{x\}$ be the fractional part of $x$, i.e. $\{x\}=x-[x]$, where $[x]$ is the biggest integer $\leq x$.
The question might be well known but I don't know where to look for: Assume $\alpha$ is an ...
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Approximation of an irrational point from a given direction
Taking norms to be maximal norm, then the simultaneous version of the Dirichlet's approximation theorem states that given real numbers ${\displaystyle \alpha _{1},\ldots ,\alpha _{d}} $,there are ...
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Density of numbers $n$ which are co-prime with their $\phi$-value
Let $n$ be a positive integer. The Euler $\phi$-function is defined by
$$\displaystyle \phi(n) = \# \{1 \leq a \leq n-1 : \gcd(a,n) = 1\}.$$
It is in fact a multiplicative function, and one has the ...
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Decide if a system of arithmetic sequences is an $m$-cover of $\mathbb{N}$
Let $A = \{ a_i + b_i \mathbb{N} \}_{i=1}^{k}$, where $a_1, \ldots, a_k \in \mathbb{N} \cup \{0\}$ and $b_1, \ldots, b_k \in \mathbb{N}$ be a system of arithmetic sequences.
For a positive integer $m$...
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gcd of polynomial values
Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...
user47305
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Vandermonde determinant: modulo
There is a lot of fascination with the Vandermonde determinant and for many good reasons and purposes. My current quest is more of number-theoretic.
QUESTION. Let $p\equiv 3$ (mod $4$) be a prime. ...
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Is there a clear-cut analogue of the strong form of Serre's Conjecture for residually reducible Galois Representations?
Let $p$ be a prime and $\mathbb{F}$ a finite field of characteristic $p$. The theorem of Khare and Wintenberger roughly states that an irreducible, odd Galois representation $\bar{\rho}:G_{\mathbb{Q}}\...
user130124
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Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)
(I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an $\text{ais}()$ already here)
...
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Does this multiplicative function have a name? If so, what is known about it?
It is well-known that the Euler $\phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $\phi(mn) = \phi(m)\phi(n)$. Thus it is defined by its values on prime powers. ...
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Irrationality of the values of the prime zeta function
Preamble: I asked this question on Math.SE with a bounty of 100, but received no replies. The bounty has now ended, and it was suggested I post this here instead.
Since Apéry we know that $\zeta(3)$, ...
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Invariant lattices of group representations over a $p$-adic field
Let $G$ be a finite group, $K$ be a $p$-adic field with an uniformizer $\pi$ and residue field $k \cong \Bbb F_q$, and $V$ be an irreducible representation of $G$ over $K$.
Let $X_{V}^G$ be the set ...
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The number of numbers no greater than n that are divisible by all their suffixes
My question: what a formula for finding the number of numbers no greater than n that are divisible by all their suffixes.
e.g: 5, 25, 125, 0125, 70125 are divisors of 70125.
refinement: $\overline{0....
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Mean value estimates for general number fields
Results are known in many different cases to bound powers of L-functions on average over a wide enough family. I am interested in results for general number fields, not only for the rationals, for ...
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Does $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converge for $\sigma > \frac{1}{2}$?
Looking at @Lucia's answer to this question it appears $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converges for $\sigma > \frac{1}{2}$. Can someone point me to a proof or provide proof for this? If I ...
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Is total degree version and $x,y$ degree version of Coppersmith's theorem correct?
The notes here https://web.eecs.umich.edu/~cpeikert/lic13/lec04.pdf have the note 'Small decryption exponent $d$: so far the best known attack recovers $d$ if it is less than $N^{.292}$. This uses a ...
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Find an integer coprime to sequence sum
Given two positive integers $a<b$, can we always find an integer $c\in [a, b]$ that is coprime to $\sum_{a\le i\le b} i$?
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The outer automorphism of the dihedral group $D_4$ and quartic polynomials
Let $L$ be a dihedral quartic field. That is, we have $[L : \mathbb{Q}] = 4$ and the Galois closure $M$ of $L$ is a degree 8 extension of $\mathbb{Q}$ with Galois group isomorphic to the dihedral ...
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Relation between Fourier coefficients and Satake parameters
Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part:
$$L(s) =...
- 5,314
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Are there Galois representations associated with any regular algebraic cuspidal automorphic representation?
Let $K$ be a number field and $\pi$ be a regular algebraic cuspidal automorphic representation on $\mathrm{GL}_2(\mathbb{A}_K)$. Let $\lambda$ be a prime of the field of Fourier coefficients of $\pi$ ...
user130124
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Lower density of numbers not summable by consecutive integers
Let us call a positive integer $n\in\mathbb{N}$ consecutively summable if there are positive integers $m, k < n$ such that $$n=\sum_{i=0}^k (m+i).$$For $A\subseteq \mathbb{N}$ we set the lower ...
- 45.5k
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Two congruence conjectures modulo prime p
How to prove the following two congruences?
Question1: Let $p\equiv 1 \pmod 3$ be a prime, then
$$\sum_{k=0\atop k\neq(p-1)/3}^{(p-1)/2}\frac{\binom{2k}k}{3k+1}\equiv 0 \pmod p.$$
...
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Remainder term in an integral linked to the Riemann zeta function
Sorry if this is not research level, but the following problem occurs in my own research: it is trivial to show that for $k\ge2$ integral we have
$\zeta(k)=(1/(k-1)!)\int_0^\infty t^{k-1}/(e^t-1)\,dt$ ...
- 11.8k
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3
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Counting configurations on a 2xn board under restrictions [closed]
Find the number of ways of selecting k cells from a $(2\times n)$-board such that no two selected cells share a side (non-adjacent).
For $n=3$ and $k=2$, the answer is $8$; for $n=5$ and $k=3$, the ...
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Dimension sum "rules" in Lie algebras [closed]
tr;dr intro: I came up with this question when I couldn't remember how many terms are in an $E_7$-ish (representing $\bigotimes$ adjoint) clebsch. Tried it on $G_2$, $7 \bigotimes 14=7+...$ argh, is ...
- 4,699
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0
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Uncertainty in semiprime factors
What is maximum $p\in(0,\frac12)$ such that if we know $2p\in(0,1)$ fraction of bits of $PQ$ with $P,Q$ primes it is possible to identify $p$ fraction of bits in each of $P,Q$ with certainty in ...
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Is a reductive group scheme always parahoric?
Let $R$ be complete (or, more generally, Henselian) discrete valuation ring with fraction field $K$. Let $G$ be a reductive $R$-group scheme. Is $G$ a parahoric in the sense of Bruhat-Tits? If so, ...
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For which primes does this iterated function act transitively? (Sort of a finite analogue of Collatz conjecture.)
Background: I was trying to prove something having to do with cyclic group actions on matroids and was able to show that what I want holds if a particular elementary-looking number-theoretic property ...
- 1,871
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1
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295
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Distribution of 'square classes' of binary quadratic forms
Let $f$ be a binary quadratic form with integer coefficients and non-zero discriminant $D$. Suppose for simplicity that $D$ is a fundamental discriminant (which in particular implies that $f$ is ...
- 25.5k
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0
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Is there a result that connects the discriminant of an order to the discriminants of the generators of an integral basis for the order?
Let $\left\{ 1 , \omega_1, \dots , \omega_{n-1} \right\}$ be an integral basis for an order $\mathcal{O}$ of a number field of degree $n$ over $\mathbb{Q}$, let $\Delta $ be the discriminant of $\...
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$\psi(2,1/6),\psi(4,1/6)$ in terms of zeta and pi only and another closed form for zeta
Let $\psi(n,x)$ denote the polygamma function.
In this answer Lucia gave linear relations for $\psi(m,1/3),\psi(m,1/6),\zeta(m+1)$.
The computer managed to find closed form for $\psi(2,1/6)$ and $\...
- 24.2k
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0
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118
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Explicit algebraic constructions of Parshin covers
Let $X$ be an algebraic curve defined over a number field $K$ and has genus $g \geq 2$. Let $P$ be a $K$-point of $X$. We say that $X_P$ is a Parshin cover of $X$ if $X_P$ is defined over a finite ...
- 25.5k
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1
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Is every prime greater than 5, less than the sum of the two previous primes?
Can one prove by elementary means (such as Paul Erdös' proof of Bertrand's Postulate) that every prime greater than 5 is less than the sum of the two primes immediately preceding it?
- 3,677
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1
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385
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$\pi$ in terms of polygamma
The computer found this, but couldn't prove it.
Let $\psi(n,x)$ denote the polygamma function.
With precision 500 decimal digits we have:
$$ \pi^2 = \frac{1}{4}(15 \psi(1, \frac13) - 3 \psi(1, \...
- 24.2k
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Is the following weak version of second Hardy-Littlewood conjecture already known?
Very recently I was going through my previous MSE posts and I stumbled upon some of them regarding the Second Hardy-Littlewood Conjecture which states that,
For all $x,y\ge 2$ we have, $$\pi(x)+\...
user57432