Skip to main content

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

92 questions from the last 30 days
Filter by
Sorted by
Tagged with
2 votes
0 answers
94 views

Is there a closed formula for factorial?

Is there a closed formula for n! that does not use the factorial notation or integrals? This question has been asked several times on math.SE but none of the answers there provides a real answer. I ...
domotorp's user avatar
  • 18.7k
0 votes
0 answers
30 views

A system of nonlinear Diophantine equations whose positive solutions are not coprime

Consider the following system of Diophantine equations: $$v_1k_1=k_1^3-k_2^3+k_3^3 \\ v_2k_2=k_1^3+k_2^3-k_3^3 \\ v_3k_3=-k_1^3+k_2^3+k_3^3 \tag{1}$$ where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
Amir's user avatar
  • 303
1 vote
0 answers
36 views

What upper bounds are known, for the number of divisors of Mersenne numbers?

Short version. What upper bounds are known, for the number of divisors of Mersenne numbers? Long version. Studying the structure of the factors of $M_n = 2^n - 1$ appears to be an active and difficult ...
Niel de Beaudrap's user avatar
3 votes
0 answers
126 views

Lemma in Roth's Theorem for Primes

I am reading Ben Green's paper Roth's Theorem in the Primes and I don't follow the proof of Lemma 6.1. I am not sure where the fact there are no more than $n^{3/4}$ elements $x\in A_0$ with $x\leq n^{...
Laurence PW's user avatar
4 votes
1 answer
501 views

Is decomposability of integer polynomials over the rational numbers an undecidable problem?

By a decomposition of a polynomial $F(x)$ over a field $K$ we mean writing $F(x)$ as $$ F(x)=G(H(x)) \quad(G(x), H(x) \in K[x]), $$ which is nontrivial if $\operatorname{deg} G(x)>1$ and $\...
SARTHAK GUPTA's user avatar
0 votes
0 answers
62 views

Hasse principle for Brauer groups of fields of transcendence degree 2

In his paper "A Hasse principle for function fields over PAC fields" (DOI link), Ido Efrat proves the following result: Let $F$ be an extension of a perfect PAC field $K$ of relative ...
aspear's user avatar
  • 31
2 votes
0 answers
95 views

Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support

This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction? Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
H A Helfgott's user avatar
  • 20.2k
7 votes
0 answers
102 views

Langlands correspondence of coverings of $\mathrm{SL}_2(\mathbb R)$ and modular forms with fractional weights

$\DeclareMathOperator\SL{SL}$Let $G \to \SL_2(\mathbb R)$ be a finite covering of degree $d \geq 2$. Then $G$ is a connected Lie group with semisimple Lie algebra $\mathfrak{g}=\mathfrak{sl}_2$ and ...
Zhiyu's user avatar
  • 6,622
0 votes
1 answer
100 views

Rational functions on elliptic curves over global fields with given support

Let $E$ be an elliptic curve over a global field $k$. Let $x_1, \dots, x_r$ be a set of generators of $E(k) / E(k)_{tor}$ (or more generally, a $\mathbb Q$-basis of $E(k)_{\mathbb Q}$), and let $x_0$ ...
yoyo's user avatar
  • 67
2 votes
0 answers
57 views

Wieferich primes and identities for the Euler quotients of $2^n+1$ and $\frac{2^n+1}{3}$

Let $n>1$ be odd integer. Define the Euler quotient $a(n)=\frac{2^{\varphi(n)}-1 \bmod n^2}{n}$. Number $n$ with $a(n)=0$ is Wieferich number and if it is prime it is Wieferich prime. It is open ...
joro's user avatar
  • 25.4k
1 vote
0 answers
100 views

The value of the Hauptmodul at CM point

Let $J$ be a classical normalized $j$-invariant (that is, J=j-744). Then, it is a classical result that $J(\tau)$ is an algebraic integer if $\tau$ is an imaginary quadratic number (that is, $a\tau^2+...
KS M's user avatar
  • 111
6 votes
1 answer
2k views

Difficulty with "A new elementary proof of the Prime Number Theorem" by Richter

I'm studying Richter's "A new elementary proof of the Prime Number Theorem" paper, and I'm finding some problems understanding some parts of it. For example, I don't see how to get, in Lemma ...
rr_math's user avatar
  • 63
3 votes
0 answers
130 views

Galois cohomology and Levi subgroups

Let $F$ a field and $G$ a smooth connected reductive group with a Levi subgroup $M$. Under what assumptions is $H^1(F, M) \to H^1(F, G)$ injective? In the case $F$ is nonarchimedean local I believe ...
C.D.'s user avatar
  • 605
0 votes
0 answers
99 views

On the form of algebraic numbers belonging to a specific field extension

Let $m>1$ be an integer and set $\theta=10^{-1/m}$. For a $\gamma\in \mathbb{Q}(\theta)$, there exists $a_0,\ldots,a_{m-1}\in \mathbb{Q}$ such that $$ \gamma=a_0+a_1\theta+\cdots+a_{m-1}\theta^{m-1}...
Jean's user avatar
  • 515
1 vote
0 answers
100 views

Curious congruences modulo $4$ involving primes

We define $$S(n)=\sum_{a=2+(n\pmod 2)}^{n-2} \sharp(\{j,1\leq j<n \pmod{a},(a,j)=1\})\ .$$ (Searching the OEIS yielded no results.) For $n>2$ we have the following experimental observations (...
Roland Bacher's user avatar
2 votes
0 answers
76 views

Function that is (essentially) a self-convolution but not a multiple of a self-convolution

Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
H A Helfgott's user avatar
  • 20.2k
1 vote
0 answers
111 views

Whether or not the root number of GL$_3\times$GL$_2$ $L$-function $L(s, F \otimes g)$ contains the coefficients $\lambda_g(n)$ of $g$?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ and $q$ be two distinct primes. Let $$\Gamma_0(p)= \left\{ g\in \GL_3(\mathbb{Z}):g \equiv \left(\begin{matrix} \ast &\ast&\...
hofnumber's user avatar
-3 votes
0 answers
70 views

Is the upper bound on $H_{1}$ a decreasing function of the proportion of critical zeros of Zeta?

This question stems from https://arxiv.org/abs/2411.19762 and the numerical observation that the best unconditional upper bound for $H_{1}:=\lim\inf_{n\to\infty}p_{n+1}-p_{n}$, namely $H_{1}^{\flat}=...
Sylvain JULIEN's user avatar
2 votes
0 answers
119 views

Polynomial discriminant equation

This is a fairly straightforward question, and I am hoping a definitive answer exists. Does there exist a quadratic form $A \in \mathbb{C}[x_1, x_2, x_3, x_4]$ and a cubic form $B \in \mathbb{C}[x_1, ...
Stanley Yao Xiao's user avatar
1 vote
0 answers
67 views

About ratio $\frac{\Omega (n)}{\omega (n)}$

What is the asymptotic estimate of $\sharp\left\{n\leq x,\frac{\Omega (n)}{\omega (n)}> 1+\varepsilon \right\}$, with fixed $\varepsilon > 0$, where $ n=s\cdot q$ ; $s$-a powerfull part of $n$ ...
Andrej Leško's user avatar
1 vote
1 answer
176 views
+50

On a probabilistic integer factorization algorithm given bounds for one prime factor

We got a probabilistic integer factorization algorithm and experimental evidence with large integers given bounds for one factor. Let $D \ge 2$ be real number and let $p,q$ be primes and $N=pq$. ...
joro's user avatar
  • 25.4k
0 votes
0 answers
92 views

The ratio $\Omega(n)/\omega(n)$ for a special set of integers

It is known that every positive integer can be expressed as $n=s\cdot q $, where $s$ is a powerful number and $q$ a squarefree number, with $(s,q)=1$. It is also known that the set of integers such ...
Andrej Leško's user avatar
2 votes
0 answers
119 views

Generalized identity with Stirling numbers of the second kind and falling factorials

It is known that Striling numbers of the second kind satisfy the relation $$ \sum\limits_{k=0}^{n}{n \brace k}(x)_k = x^n. $$ where $(x)_n$ is the falling factorials such that $$ (x)_n = x(x-1)(x-2)\...
Notamathematician's user avatar
-3 votes
0 answers
140 views

Divisors of n and n + 1

Suppose $a$ is a proper divisor of $n$ (where $n$ is a positive integer), and $b$ a proper divisor of $n + 1$. Is there a general criterion (or general property of $n$) which enables one to conclude ...
THC's user avatar
  • 4,547
5 votes
1 answer
469 views

Is the set of generalized Fermat triples computable?

Is $\;\big\{(a,b,c)\in\mathbb{N}^3: \big(\exists m,n,\ell \in (\mathbb{N}\setminus\{0,1,2\})\big): a^m + b^n= c^\ell\big\}\;$ computable?
Dominic van der Zypen's user avatar
2 votes
0 answers
93 views

Galois representations attached to discrete automorphic representations

Let $F$ be a totally real field. Let $G$ be a (split) connected reductive group over $F$. Let $\pi$ be an irreducible automorphic representation of $G$. Recall in the work of Buzzard and Gee "The ...
Zhiyu's user avatar
  • 6,622
1 vote
0 answers
82 views

Descent of isogenies between p-divisible groups

Let $\mathcal{G}$ be a $p$-divisible group over $K$, which is a finite extension of $\mathbb{Q}_p$. Let $\rho: \text{Gal}(\bar{K}/K)\rightarrow \text{GL}(T_p\mathcal{G})$ be the associated Galois ...
Kris's user avatar
  • 11
1 vote
0 answers
263 views

Are there connections between Calabi-Yau manifolds and number theory?

I am interested in understanding whether there are any significant connections between Calabi-Yau manifolds and number theory. Calabi-Yau manifolds are central objects in algebraic geometry and string ...
Abdullah M Al-jazy's user avatar
1 vote
0 answers
29 views

Factoring semiprimes via sum of two squares? [migrated]

The following thoughts came into my head after watching Grant Sanderson's JBPM award lecture here, in which he discusses the fact that we can quickly factor 3599 by noticing it can be written as (60-1)...
weissguy's user avatar
16 votes
2 answers
2k views

Polynomials for natural numbers and irreducible polynomials for prime numbers?

Let $p$ be a prime and $n$ be a natural number. Define inductively for prime numbers: $f_1(x) := 1$, $f_2(x):=x$, $f_p(x) := 1+\prod_{q\mid p-1} f_q(x)^{v_q(p-1)}$. Is $f_p(x)$ always irreducible for ...
mathoverflowUser's user avatar
2 votes
1 answer
122 views

Questions about elliptic curves with level-$n$ structure

Let $n$ be a positive integer, which is $4$ or a prime number $l$. Let $E$ be an elliptic curve defined over a number field $K$. Assume that all the $n$-torsion points of $E$ are defined over $K$, i.e....
yoyo's user avatar
  • 67
4 votes
0 answers
87 views

Levis, parabolics and Bruhat-Tits over Henselian local rings

Let $(R,m)$ be a Henselian local ring with algebraically closed or finite residue field $k$ and fraction field $F$. For example, we may work with $R=W(\mathbb F_p^{alg})$. The paper "Reductive ...
Zhiyu's user avatar
  • 6,622
30 votes
2 answers
1k views

Why does this system of gcd equations have no solutions?

In February 2024 the following question was posed by user @Aig on Math.StackExchange: Find the solutions to the system of equations $$\begin{cases} a + b = \gcd(a^3,b^3) \\ b+c = \gcd(b^3,c^3) \\ c+a =...
Sahaj's user avatar
  • 351
3 votes
0 answers
77 views

primes that ramify in division fields for hyperelliptic jacobians

Let $C$ be a hyperelliptic curve $y^2=f(x)$ of genus $g\geq 2$ and $\Delta$ the discriminant of $f(x)$. Let $\ell>2$ be a prime that divides $\Delta$ to the order $e:=\operatorname{ord}_\ell(\Delta)...
Anwesh Ray's user avatar
3 votes
0 answers
239 views

Multiplicative functions and Dirichlet characters

I am studying Dirichlet characters and modular functions as part of my research, specifically working through concepts found in the paper "An explicit hybrid estimate for $L(1/2 + it)$" by ...
Fatima Majeed's user avatar
1 vote
1 answer
186 views

Existence of Finite Amicable Groups

I'm interested in exploring the concept of "amicable groups" as follows: Definition. Two finite groups $G$ and $H$ are called amicable groups if: $G$ is the direct sum of proper subgroups ...
Maziar Esfahanian's user avatar
-1 votes
0 answers
49 views

$s^2 + 1$ has no prime factors $p \equiv 3 \mod 4$? [migrated]

$s^2 + 1$ is the squared length of a vector in a square lattice for every integer $s$. The number of integer solutions in $a, b$ for $a^2 + b^2 = s^2 + 1$ is of course $> 0$, e.g. $a = s, b = 1$ is ...
Helmut Ruhland's user avatar
0 votes
0 answers
118 views

Uncomplete argument in Nishioka book

In Nishioka book "Mahler functions and transcendence" in the proof of Theorem 4.2.1, Nishioka asserts the following: For a matrix $A=(a_{i,j})_{1\le i\le m}$ with coefficients in $K[z]$ ($K$ ...
joaopa's user avatar
  • 3,996
4 votes
0 answers
513 views

Deriving inequality (8.9) from (8.8), in Iwaniec–Kowalski “Analytic Number Theory”

I am working through the problem presented in Chapter 8 of Iwaniec and Kowalski’s Analytic Number Theory (specifically inequalities (8.8) and (8.9)) and I am struggling with the transition between ...
Fatima Majeed's user avatar
0 votes
1 answer
141 views

A lower bound for the largest prime divisor of an integer

I have often heard it stated that Erdős conjectured the following: For any integer $n > 1$, there exists a prime divisor $p$ of $n$ such that $$p > c \cdot \log \log n,$$ where $c > 0$ is a ...
MAY's user avatar
  • 55
1 vote
0 answers
116 views

Can all congruences for a third-order recurrence relation hold for some composite $n$?

Let $p$ be a prime with $p \gt 3$. Consider the polynomial $f = x^3 - 3x -1$. Suppose $f$ is irreducible over $\mathbb{F}_{p}$. Let $E$ be the splitting field of $f$ over $\mathbb{F}_{p}$, and let $\...
David Bernier's user avatar
-1 votes
2 answers
371 views

Are these polynomials the same? [closed]

Let $p=2^{127}-1,P,Q\in \mathbb F_p[x]$ with $P(x)=x^2+1$ and $Q(x)=x^2+2$. Are there some polynomial $H \in \mathbb F_p[x]$ bijectif on $\mathbb F_p$ with $\forall x \in \mathbb F_p, H(P(x))=Q(H(x))...
Dattier's user avatar
  • 4,074
0 votes
0 answers
126 views

Asymptotics of sum $\sum_{d \mid n} \frac{\mu(d)}{d^2}$

Is anything known about the asymptotics of the sum $$ \sum_{d \mid n} \frac{\mu(d)}{d^2} $$ as $n$ tends to infinity? I am particularly interested in $\liminf_{n \to \infty}$. Here, $\mu$ is the ...
darko's user avatar
  • 309
2 votes
1 answer
126 views

Changing the weight space for an eigenvariety

Let $G$ be an algebraic group (like $\operatorname{GL}_2$ or $\operatorname{GL}_2 \times\operatorname{GL}_2$ for example). Assume that there exists an eigenvariety $\mathcal{E}^G$ parameterizing ...
BanAna's user avatar
  • 93
0 votes
1 answer
129 views

Closed form for $\sum\limits_{k=0}^{n} [\operatorname{wt}(k) = m]$ where $\operatorname{wt}(n)$ is the binary weight of $n$

Let $\operatorname{wt}(n)$ be A000120 (i.e., number of $1$'s in binary expansion of $n$). Let $a(n,m)$ be the family of integer sequences such that $$ a(n,m) = \sum\limits_{k=0}^{n} [\operatorname{wt}(...
Notamathematician's user avatar
0 votes
0 answers
86 views

How to prove the following equation (involving multiple binomial coefficients sum)?

I encountered the equation below, encountered a problem that has been bothering me for a long time Does anyone have an idea how to prove it? I would be extremely grateful to you if you come up with an ...
tongjun's user avatar
  • 41
2 votes
0 answers
194 views

Functions such that the *integral* of the Fourier transform is non-negative?

Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that $$\int_{-\infty}^x \widehat{f}(t) dt \...
H A Helfgott's user avatar
  • 20.2k
3 votes
0 answers
190 views

Voronoi formula on $\mathrm{GL}_4$ in the level aspect with ramification

$\DeclareMathOperator\GL{GL}$Let $f$ be an automorphic form on $\GL_3$ for $\Gamma_0(p)$ with $p$ being a prime (see Bump or Goldfeld's books for definitions). Recall that, in this paper-"The ...
hofnumber's user avatar
8 votes
1 answer
670 views

Infinite series and sum of two squares

Consider the following infinite sequence $a(n)$ generated by $$\sum_{n\geq0} a(n)q^n =\frac{\sum_{k\geq0}F(2k+1)q^{\binom{k+1}2}}{\sum_{k\geq0} q^{\binom{k+1}2}}$$ where the $F(2k+1)$ are the odd ...
T. Amdeberhan's user avatar
3 votes
0 answers
63 views

Congruences regarding $4n$-dimensional lattices

A sequence of integers $(a_n)_{n\geq 1}$ satisfies Gauss congruence if $$\sum_{d\mid n}\mu(d)a_{n/d}\equiv 0\pmod{n}$$ for every $n\geq 1$. Such sequences are also called Dold sequences, Newton ...
fern-gossow's user avatar