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
14,727
questions
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Infinite tamely ramified $p$-extensions of $\mathbb{Q}$ contain infinite unramified subextensions?
Let $p$ be a prime. By a $ p $-extension we mean a Galois extension whose Galois group is a $ p $-group. Let $L$ be an infinite tamely ramified $p$-extension of $\mathbb{Q}$, i.e. all primes ramified ...
- 308
3
votes
1
answer
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Conductor at 2 of abelian surfaces with real multiplication
Let $A/\mathbb{Q}$ be an abelian surface such that $\text{End}_{\mathbb{Q}}(A)\otimes \mathbb{Q}$ is a real quadratic field $E$. I am interested in bounding the conductor of $A$ at $2$.
Let $\mathfrak{...
- 1,262
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21
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If $p^k m^2$ is an odd perfect number, then $\sigma(m^2)/p^k \equiv \sigma(p^k)/2 \pmod 8$. Can you then rule out $\sigma(m^2)/p^k = \sigma(p^k)/2$?
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
A number $M$ is called perfect if it satisfies $\sigma(M)=2M$.
Euler proved that a hypothetical odd perfect ...
- 1,334
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Is there a "convolution" of asymptotic growth?
Suppose that I have two asymptotic counts given by
$$
\#\{x \in [0,H] \cap \mathbb Z: f(x) \leq H\} \sim F(H)
$$
and also
$$
\#\{x \in [0,H] \cap \mathbb Z: g(x) \leq H\} \sim G(H).
$$
From these two ...
- 41
3
votes
0
answers
45
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On Carmichael function and aliquot parts of odd perfect numbers
I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
- 89
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Partial exponential sums over lattice points of lattice cones
Consider the usual lattice $M:=\mathbb{Z}^2\subseteq\mathbb{R}^2$, and
let $v_1,v_2\in\mathbb{Z}^2\subseteq\mathbb{R}^2$ be two non-zero lattice points which are $\mathbb{Z}$-linearly independent. ...
- 7,341
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164
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Why is a Galois group of a finite Galois extension of global fields K/F transitive on the set of primes of K lying above P, a prime of F? [migrated]
Let $K/F$ be a finite Galois extension of global extensions, and let $P$ be a prime of $F$. Letting $S$ be
$$
\{Q : Q \ \text{is a prime of} \ K \ \text{that lies above} \ P\}
$$
show that $G={\rm ...
- 17
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1
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Fourier series of Eisenstein series — elegant and very good approximation
I played around with the Fourier series of the Eisenstein series resp. divisor sums and did some calculations, see below. Although the deduction is not rigorous / wrong (as the power series for the ...
- 376
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Lucas–Lehmer test and triangle of coefficients of Chebyshev's
In the Lucas–Lehmer test with $ \quad p \quad $ an odd prime.
we know that $ \quad S_0=4 \quad $ and $ \quad S_i=S_{i-1}^2-2 \quad $ for $\quad i>0 \quad$
$M_p=2^p-1 \quad$ is prime if $ \quad S_{p-...
- 159
1
vote
1
answer
243
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Sheaf cohomology in number theory
I have read the first three chapters of Hartshorne and was wondering what are the applications of the notions presented in number theory or arithmetic geometry. I already know that the notion of ...
- 123
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votes
1
answer
114
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Epstein zeta function of Barnes-Wall and related lattices
Sarnak and Strömbergsson studied Epstein zeta function $\zeta(L,s)=\sum\limits_{0\neq v\in L}\langle v,v\rangle^{-s}$ of a number of highly symmetric lattices in their Inventiones Math. paper.
In ...
- 13.2k
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61
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Clumps of small multiples of large squares
Am I right to be surprised by this big clump of numbers divisible by large squares within a not-so-long interval? If so, should I be surprised because $(1)$ this rarely happens, or because $(2)$ it's ...
- 10.2k
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votes
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Existence of intermediate field extensions for tamely ramified p-adic extensions
Let $p$ be a prime, and let $K/\mathbb{Q}_p$ be a tamely ramified finite extension of degree $n$. Let $q$ be a prime factor of $n$ with $q\neq p$. Must there exist an intermediate extension $L$ (...
- 421
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About simple motives
I'm reading through Jannsen's paper Motives, numerical equivalence, and semi-simplicity and I'd like to pose two questions.
Suppose all motives are $F$-linear, for some characteristic zero field $F$, ...
- 13
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answers
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Triples of integers a, b and c with a + b = c and specified prime divisors
Given an integer $n$, let $P(n)$ denote the set of odd prime divisors of $n$.
Let $\Delta$ be the simplicial complex over the set of sets of odd prime numbers
which consists of the simplices $S$ such ...
- 18.5k
1
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0
answers
62
views
Show that a region in a plane defined by a polynomial contains integer points
Let $F \in \mathbb{Z}[x,y]$ be a polynomial of degree $2n$ such that the homogeneous degree $2n$ part of $F$, say $F_{2n}$, is positive semi-definite. How does one show that for some $\delta_n > 0$ ...
- 22.5k
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Research of average number of equivalence classes of solutions to generalised Pell's equation
Statement of the problem
Firstly, consider the infamous Pell's equation:
$x^{2}-dy^{2}=1$. Here $x$ and $y$ are integers and $d$ is a nonsquare integer. It is known ([3]) that all solutions of this ...
- 31
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votes
1
answer
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Using the Lehmer quintic to solve $11$-degree equations and higher?
(This is a natural continuation of a previous post.)
I. Quintic method
Given the Lehmer quintic,
$$x^5 + n^2x^4 - (2n^3 + 6n^2 + 10n + 10)x^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)x^2 + (n^3 + 4n^2 + 10n + ...
- 10.3k
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235
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Is $\sum\limits_{k=0}^{\infty}\frac{1}{(mk)!^{n+1}}$ irrational?
I was using Wolfram Alpha for things, and I came across $I_{0}(2)$. For fun I tried asking Wolfram Alpha if the number was irrational, but said it's unknown. I believe this is an error, as its ...
- 101
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300
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On the OEIS sequence A327265
The OEIS sequence https://oeis.org/A327265 starts:
$$1, 2, 5, 11, 19, 31, 51, 89, 123, 151, 179, 181, 180, 365, 634, 657, 656, 655.$$
$A327265(n)$ is the smallest $k$ such that A309981$(k) = n$.
$...
9
votes
1
answer
490
views
Baker's theorem for integer combinations of logarithms of integers?
Baker's theorem in transcendental number theory states that
$$
\left|\beta_0 + \sum_{i=1}^n \beta_i \log \alpha_i\right| > H^{-C}
$$
where
$\beta_0, \ldots, \beta_n$ are algebraic numbers, not ...
- 846
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votes
0
answers
74
views
Eta product of squared tau function
The Ramanujan tau function is the coefficient of the 24th power of the Dedekind eta function.
$$ \eta(x)^{24}= x\prod_{m=1}^\infty (1 - x^m)^{24} =\sum_{n=1}^\infty \tau(n)\,x^n , $$
I want to know ...
- 363
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0
answers
88
views
Can we tweak the Möbius function sum to better converge on the critical line and maybe also to the left of it?
Let the constant $c = -3/4$ and let the usual divisibility matrix $B(n,k)=1$ if $k\mid n$ else $B(n,k)=0$ for all integers $n \geq 1$ and $k \geq 1$
and let the matrix $A$ be: $$A=B-I(1+c)$$
where $I$ ...
- 1,053
4
votes
1
answer
320
views
Primes of the form $d^2+d+1$
Is $d^2+d+1$ prime for infinitely many $d\in \mathbb{Z}_{>0}$?
This is expected by the Bunyakovsky conjecture which says that, under some conditions, given a polynomial $p(x) \in \mathbb{Z}[x]$ we ...
- 348
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votes
1
answer
293
views
Generating prime $\ p_{n+1}\ $ (the complete version)
Let $\ p_n\ $ be the consecutive primes starting with
$\ p_0:=2.\ $ Let $\ M(n)\ $ be the multiplicative monomial
generated by $\ \{p_k:\ k=0\ldots n\}\ $ (of course $\ 1\in M(n)$).
Could you prove or ...
- 4,172
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answers
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views
What is the conductor of $K(\sqrt{2})$ over $K$?
Let $ K=\Bbb Q\left(\sqrt{(-1)^\frac{N+1}{2}N}\right)$. I want to find the ray class field of $K$ containing $\sqrt{2}$. I considered $L=\Bbb Q(\sqrt{2})$. By Artin reciprocity, there exist modulus $\...
- 673
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views
A similar relationship between the generic cubic and the Lehmer quintic?
I. Comparison
It doesn't seem to be well-known that the generic cubic (prominent in this MO post) for $C_3 = A_3$,
$$x^3-nx^2+(n-3)x+1 = 0$$
has the nice property that its roots $a,b,c$, if in correct ...
- 10.3k
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votes
0
answers
80
views
Evaluation of mock modular forms at elliptic points
The holomorphic function
$$F(\tau)=-\frac{1}{\vartheta_4(\tau)}\sum_{n\in\mathbb Z}\frac{(-1)^nq^{\frac{n^2}{2}-\frac 18}}{1-q^{n-\frac12}}=2q^{\frac38}(1+3q^{\frac12}+7q+14q^{\frac32}+\dots),$$
is a ...
- 67
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votes
2
answers
479
views
Reference request for recurrence relation of division polynomials
The recurrence relations for division polynomials of elliptic curves are well known:
$$\Psi_{2n} = \Psi_n \left( \Psi_{n+2} \Psi_{n-1}^2 - \Psi_{n-2} \Psi_{n+1}^2 \right) / \ 2y$$
$$\Psi_{2n+1} = \...
- 265
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votes
2
answers
342
views
A method to generate solvable equations of degrees $p = 7, 13, 19, 31, 37,\dots$ using only cubics
I've always wondered if the DeMoivre method to generate an algebraic number $x_p$,
$$x_p = u_1^{1/p}+u_2^{1/p}$$
of degree $p$ using only quadratic roots $u_i$ could be generalized using cubic roots $...
- 10.3k
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votes
0
answers
141
views
Remainder-balancedness of primes
Let $\mathbb{N}_+$ denote the set of positive integers. Consider the remainder function $\text{rem}:\mathbb{N}_+\times \mathbb{N}_+ \to \mathbb{N}\cup\{0\}$ defined by $$(n,d) \mapsto n - \Big(\Big\...
- 41.4k
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votes
2
answers
642
views
Integer solutions of an algebraic equation
I'm trying to find integer solutions $(a,b,c)$ of the following algebraic equation with additional conditions $b>a>0$, $c>0$.
$(-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2) + 2 a b (-a^2+b^2+c^2)(...
- 651
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votes
1
answer
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views
Convergence of scrambled product for Dirichlet-$L$ function with modulo 4 character
A Dirichlet-$L$ function is typically defined by its series, and its Euler product is a consequence of the definition. Here my approach is the other way around.
I define the function
$$
L_4^*(s) = \...
- 2,690
2
votes
0
answers
157
views
Existence of fully supported element in a finite-dimensional vector space over $\mathbb{F}_p$ (and in finite abelian groups)
Let $V$ be an $n$-dimensional vector space over $\mathbb{F} = \mathbb{Z} / (p)$, the field of $p$ elements, $p$ a prime, with $\{v_1, \dotsc, v_n \}$ a basis for $V$. An element $x \in V$ is called &...
- 151
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votes
1
answer
362
views
Langlands-Shahidi method in classical language
The Langlands-Shahidi method says that the $L$-functions of automorphic representations appear in the constant terms of Eisenstein series. Since those Eisenstein series have analytic continuation and ...
- 1,223
4
votes
0
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107
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Weak version of (elliptic analog) Artin's primitive roots conjecture
Let $E/\mathbb{Q}$ be an elliptic curve, and $P\in E(\mathbb{Q})$ be any non-torsion point. Given any $\varepsilon>0,$ how often it is true that $\mathrm{ord}(P \pmod p)>p^{1-\varepsilon},~p~\...
- 471
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votes
1
answer
70
views
A sum related to the first moment of quadratic $L$-functions at $s=1$
Let $(\frac{m}{n})$ be the Jacobi quadratic symbol defined for positive squarefree odd integers $n,m$. Does the following sum go to infinity?
$$
\sum_{1\leq n \leq (\log x)^{100} } \mu^2(2n) \sum_{(\...
- 2,735
3
votes
1
answer
58
views
Frobenius-Schur indicator of a self-dual L-parameter
Let $F$ be a non-archimedean field and let $\pi$ be a self-dual supercuspidal representation of $\mathrm{GL}_n(F)$ (which, by a result of Adler exists only when $n=1$ or $n$ is even). Then, under LLC ...
- 343
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votes
0
answers
143
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A detail in I. Piatetski-Shapiro and S. Rallis's "Doubling paper": computing the integral on negligible orbits
I'm currently reading the paper "L-functions for the classical groups" by I. Piatetski-Shapiro and S. Rallis, where they introduced the doubling method over classical groups.
I'm confused at ...
- 271
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146
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Completing some of Ramanujan's results on $p = 5, 7, 9, 13, 25$?
This gathers scattered results together to see if they can be extended. The question is at the end. Given the Ramanujan theta function $f(a,b)$ and define Ramanujan's theta ratio formula,
$$r_k = (-1)^...
- 10.3k
3
votes
1
answer
179
views
On the refined minimal ramification problem for $p$-groups
Let $p$ be a prime. The minimal ramification problem is to ask whether or not every finite $p$-group $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ with exactly ...
- 308
1
vote
0
answers
79
views
Subsequence such that $c(a(n))=2^n$
Let $a(n)$ be A060831, i.e., $\sum\limits_{k=1}^{n}\operatorname{number of odd divisors of} k$.
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
Let
$$b(n,k)=2b(n,k-1)-2^{k-1}, b(n,0)=n$$
Let $c(n)$ ...
- 1,887
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votes
0
answers
168
views
Are $\pi$ and $e^{\pi i \sqrt{d}}$ algebraically independent?
Let $-d\in\mathbb{N}$ be square-free. Nesterenko (https://doi.org/10.1070/SM1996v187n09ABEH000158) proved that $e^{\pi\sqrt{-d}}$ and $\pi$ are algebraically independent. Is it known whether $e^{\pi\...
- 141
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votes
2
answers
312
views
On V. Arnold's trinities regarding PSL(2,5), PSL(2,7), and PSL(2,11)?
Given the Ramanujan theta function,
$$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$
Let $q = e^{2\pi i \tau}$ and assume $\tau = \sqrt{-d}.$ Then the following functions for levels $...
- 10.3k
0
votes
1
answer
75
views
Sums of powers of measures of $p$-adic balls
Let $(a_n,k_n) \in \mathbb{Z}_p \times \mathbb{N}$ for $n \in \mathbb{N}$ and consider the sequence of closed $p$-adic balls $B(a_n,k_n) = a_n + p^{k_n}\mathbb{Z}_p$. I assume that the $(a_n,k_n)$ are ...
- 19k
8
votes
0
answers
129
views
Explicit constructions of Ramanujan graphs
I am trying to find a list of all the explicit constructions of Ramanujan graphs. By a Ramanujan graph I mean a $k$-regular multi-graph $G$ such that all the non-trivial eigenvalues $\lambda$ of the ...
- 235
5
votes
0
answers
432
views
On the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties"
I am trying to understand section (3) of the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties" in detail. In particular, there is the following sentence on page ...
- 83
4
votes
2
answers
441
views
On the equation $a^4+b^4+c^4=2d^4$ in positive integers $a\lt b\lt c$ such that $a+b\ne c$
Background: The equation
$$a^4+b^4+c^4=2d^4$$
has infinitely many positive integral solutions if we take $c=a+b$ and $a^2+ab+b^2=d^2$ further assuming that $GCD(a,b,c)=1$.
Main problem: Find some ...
5
votes
1
answer
207
views
Relation between $G_{\mathbb{Q}_p}$ for different primes
Let $G_{\mathbb{Q}_p}$ denote the absolute Galois group of the $p$-adic field $\mathbb{Q}_{p}$. Also, their structure as abstract groups is completely known.
It is well known that this group embeds ...
- 429
5
votes
1
answer
396
views
Discrete log problem modified
Suppose one is given an odd prime $p$, a generator $g$ of $(\mathbb Z/p \mathbb Z)^*$ and two integers $a$ and $b$. Is there an efficient method to determine whether $\log_g a < \log_g b$? (Here we ...
- 2,381