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
4,673
questions with no upvoted or accepted answers
14
votes
0
answers
1k
views
Lifting Abelian Varieties to p-adic fields
Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...
14
votes
0
answers
463
views
Cohomological interpretations of quadratic form invariants over rings?
The standard approach to classifying of quadratic forms over $\Bbb Q$ is to use the Hasse (local-global) principle together with a system of standard invariants of quadratic forms over the local ...
14
votes
0
answers
807
views
Does Fermat hold in non-standard models?
Let $n \geq 3$ be a natural number and $PA$ denote Peano arithmetic. Do we have
$PA \models \forall x,y,z \geq 1 : x^n + y^n \neq z^n$?
In other words, does Fermat's Last Theorem hold also in non-...
13
votes
0
answers
305
views
Upper bound on prime powers in interval
I just spent a full day on the brutish and thankless task of proving that the Brun-Titchmarsh bound holds for prime powers (including primes), and not just for primes, in the following senses:
(a) the ...
13
votes
0
answers
644
views
List of problems that Erdős offered money for?
Is there a list somewhere of all the problems that Erdős offered cash awards for, including both solved and unsolved problems? One would think that the answer is yes, but so far I have had no luck ...
13
votes
0
answers
212
views
A game based on the Euclidean algorithm
The following game is based on a somewhat "stupid" version of the Euclidean algorithm (where we allow only subtractions).
Positions are given by finite non-empty multisets (repeated elements ...
13
votes
0
answers
310
views
Work of Atkin on the 26th power of eta
The 26th power of the Dedekind $\eta$ function has been mentioned several times here on MO:
A 14th and 26th-power Dedekind eta function identity?
What's the status of the following relationship ...
13
votes
0
answers
314
views
How do we deduce the Jacquet-Langlands correspondence from Fargues' two towers?
In trying to understand the geometric proof of the local-Langlands and Jacquet-Langlands correspondence which uses Fargues's two tower theorem, I am having trouble finding a nice source on this, and I ...
13
votes
0
answers
591
views
On a kind of Hilbert irreducibility theorem
Let us work over a number field $k$. Let $C$ be a non-empty open subscheme of $\mathbb{P}^{1}_{k}$, and $X\to C$ a family of smooth, projective hyperbolic curves such that $X(k)\to C(k)$ is surjective....
13
votes
0
answers
403
views
Is this arithmetic function strictly positive and unbounded?
As requested by Mathphile, since there have been efforts but no complete solutions to some questions raised when this question was asked on MSE, and since we think that here the question is more ...
13
votes
0
answers
1k
views
Why am I unable to find primes of the form $(9n)!+n!+1$?
See also Math StackExchange: Is there a prime of the form $(9n)!+n!+1$?
Recently, user Peter from Math StackExchange asked for a prime of the form $(9n)!+n!+1$ (where $n$ is some natural number).
...
13
votes
0
answers
506
views
When does the product equal the sum?
Let $R$ be a commutative ring with identity and $R^n$ be the direct sum of $R$. Find all $a_1, a_2, \cdots, a_n \in R$ such that $$a_1 + a_2 + \cdots + a_n = a_1a_2\cdots a_n,$$
or, in other words, if ...
13
votes
0
answers
197
views
Hypergeometric representation of Eisenstein series
It is well known (Fricke ?) that $E_4^{1/4}$ and $E_6^{1/6}$ can be represented as Gauss hypergeometric functions of $1728/j$ and $1728/(1728-j)$ respectively.
The same result is true in levels $2$, $...
13
votes
0
answers
488
views
Hensel lemma and rational points in complete noetherian local ring
Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.
If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...
13
votes
0
answers
345
views
A determinant problem for primes $p\equiv 1\pmod4$
Let $p$ be an odd prime, and let $A_p$ denote the matrix
$$[a_{ij}]_{1\le i,j\le (p-1)/2},$$
where
$$a_{1j}=\left(\frac jp\right),\ \ \text{and}\ \ a_{ij}=\left(\frac{i^2+j^2}p\right)\ \text{for}\ i&...
13
votes
0
answers
356
views
Lifting automorphic Galois representations to arithmetic fundamental groups and their quotients
Suppose $V$ is an algebraic variety over a number field $K$. The absolute Galois group $G_K$ of $K$ acts by outer automorphisms on the étale fundamental group $\pi_1(V_{\bar{K}})$ where $\bar{K}$ is ...
13
votes
0
answers
447
views
The trace formula over function fields
There are many examples in number theory where an "arithmetic" problem (i.e. for number fields) has an easier analogue for function fields over finite fields. This is also true for questions ...
13
votes
0
answers
537
views
Case D=4l in Elkies' paper on Supersingular Primes of an Elliptic Curve over $\mathbb{Q}$
My question is regarding Elkies' paper on "The existence of infinitely many supersingular primes for every elliptic curve over $\mathbb{Q}$".
In the section "Nuts and Bolts", Elkies has the ...
13
votes
0
answers
471
views
show that there exist $n$ such that $r|\binom{p^n}{q^n}$
Cross-Posted from Math Stackexchange
Two positive integers $p,q$ and a prime $r$ are given, such that $r>p>q>1$. I have to show that there exist $n$ such that
$$r|\binom{p^n}{q^n}$$
Should ...
13
votes
0
answers
257
views
Group homology $\mathrm{SL}_2$ acting on $\mathrm{Sym}^g$
Let $k$ be a field. We write $\mathrm{Sym}^g(k^2)$ for the $g$-th symmetric power of the (a?) standard representation of $\mathrm{GL}_2(k)$ ($g\geq 0$ an integer). Here I consider $\mathrm{Sym}^g(k^2)$...
13
votes
0
answers
659
views
Uniform proof of Hasse principle for algebraic groups?
Let $G$ be a simply connected semi-simple linear algebraic group over a global field $k$. The Hasse principle for algebraic groups states that the map $$H^1(k,G)\rightarrow\prod_vH^1(k_v,G)$$ is ...
13
votes
0
answers
611
views
No Siegel-Landau zeros for $\mathrm{GL}(n)$
The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact:
There ...
13
votes
0
answers
282
views
Propagation of modularity and the Artin conjecture
The (still incomplete) solution of the Artin conjecture on dimension $\leq2$ has been a massive research effort that has spanned (knowingly or not) around a century.
A very natural question is, what ...
13
votes
0
answers
424
views
Must the sum of the digits of $n^k$ decrease infinitely often, for $n,k\in\mathbb{N}$ and $n$ not a power of $10$?
This is a (rephrased) repost of a question I asked on MSE about 6 months ago, but didn't receive a definitive answer.
Let $S(n)$ be the sum of the digits of $n$ (in base $10$), is it true that for ...
13
votes
0
answers
449
views
Are there infinitely many N^3 (especially for prime N) that cannot be expressed as a sum of three positive cubes?
The sequence A023042 on the OEIS website shows that a large percentage of $N^3$ are a sum of three positive cubes. OEIS lists only N<1770, but we can extend that:$$\begin{array}{|c|c|}
N&\text{%...
13
votes
0
answers
635
views
Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$
About a month ago, I got the following :
For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that
$$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{...
13
votes
0
answers
296
views
Is the set of numbers $\{ [n^{3/2}] \mid n\text{ an integer}\}$ a basis of order 3?
A set of integers is called a basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of the set.(Nathanson, Additive Number Theory, page 7)
...
13
votes
0
answers
543
views
If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$?
Question : For every even $k\ge 4$, is the following $(\star)$ true?
$$\begin{align}\text{If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$.}\...
13
votes
0
answers
407
views
Integer-valued power towers
$$\text{Let }f_n(a)=\underbrace{2^{2^{.^{.^{.^{2^a}}}}}}_{\text{$n$ 2s}}.$$
Obviously, $f_n(a)$ is an integer for every positive integer $n$ and non-negative integer $a$.
Are there any positive ...
13
votes
0
answers
582
views
Should the number of small solutions in Roth's theorem be bounded uniformly, assuming the target is an algebraic integer?
Consider, on the one hand, algebraic integers $\alpha$ and their rational approximants to within a varying exponent $\kappa > 2$; and on the other hand, smooth projective geometrically irreducible ...
13
votes
0
answers
2k
views
Why doesn't functoriality immediately imply the modularity theorem?
Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. ...
13
votes
0
answers
594
views
What's about N. M. Katz' "over-world" of exp. sums?
Having just read in N. M. Katz' beautiful old survey on exponential sums a d differential eq.s, I wonder what became out of his question (on p. 297 - 300) on a "general conceptual framework in which ...
13
votes
0
answers
548
views
Higher-dimensional algebraic subgroups of the proalgebraic Nottingham group?
Let $R$ be a commutative ring, and, for $n\ge0$,
${\mathcal{A}}_n={\mathcal{A}}_n(R)$ the group of series
$u(x)=\sum_0^\infty a_jx^{j+1}\in R[[x]]$ for which
$a_0\in R^\times$ and $u(x)\equiv x\pmod{x^...
13
votes
0
answers
710
views
Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable $\mod 4$?
Is A001935 Number of partitions with no even part repeated efficiently computable $\mod 4$?
I am interested because of this relation with sum of divisors of $8n+1$.
$\sigma(8n+1) \equiv A001935(n) \...
13
votes
0
answers
1k
views
Stickelberger's congruence for Gauss sums
We begin with (unfortunately, quite a bit of) notation necessary to state Stickelberger's congruence:
Fix an integer $k>1$, and let $p$ be a prime number not dividing $k$. Let $r$ be the smallest ...
13
votes
0
answers
703
views
Regular languages of matrices and their generating functions
My question is somewhat related to this question.
Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...
13
votes
0
answers
1k
views
A question about Mobius inversion
I don't know how precise I can make this question. I want to know whether there is a theorem that says that a certain phenomenon always happens, but I think the best I can do in order to pin down the ...
13
votes
0
answers
1k
views
Effective proofs of Siegel's theorem using arithmetic geometry
This is a speculation and perhaps naive. The theorem of Siegel that
There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is a ...
13
votes
0
answers
765
views
A conjecture on p-divisibility of Bernoulli numbers
Is anyone aware of the history of the following conjecture on the $p$-divisibility of (the numerators) of Bernoulli numbers?
CONJECTURE: For $p$ an odd prime, and $k$ even with $2 \leq k \leq p-3$, $...
12
votes
0
answers
418
views
Is there a mistake in Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions"?
I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the ...
12
votes
0
answers
447
views
Around the Erdős-Ginzburg-Ziv theorem
(Here is a problem that emerged in a conversation with Fedor Petrov and should really be a sort of "joint posting" if this format were supported.)
For any positive integers $k_1\le k_2\le\...
12
votes
0
answers
656
views
Kihara-like Z/6Z elliptic curve families
Shoichi Kihara constructed a family of elliptic curves with Mordell–Weil group $\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}^3$ (generic rank at least 3) in 2006. Kihara's family produces a number of rank 8 ...
12
votes
0
answers
523
views
When does Matiyasevich's theorem "kick in"?
Hilbert's 10th problem was famously resolved by the Matiyasevich–Robinson–Davis–Putnam theorem: the theorem implies that there is no algorithm which decides whether a given polynomial equation with ...
12
votes
0
answers
538
views
On a revised quantum Riemann hypothesis
This post provides a revision of the disproved quantum Riemann hypothesis proposed 2 years ago in this post, where you can refer to have more details about the motivations, the notations and the ...
12
votes
0
answers
260
views
Mixed characteristic analogue of algebraicity of the diagonal of two-variable power series?
Let $f=\sum_{n,m \geq 0}^{\infty}[a_{nm}]p^ny^m \in \mathbb Z_p[[y]]$, where $a_{nm} \in \mathbb F_p$ and $[\cdot]$ means the Teichmüller lifting. Define $I(f)=\sum_{n \geq 0}[a_{nn}]p^nt^n \in \...
12
votes
0
answers
257
views
sequences in non-abelian group cohomology
In general, if we have a (pro-)finite group $G$ and a sequence of (continuous) non-abelian $G$-modules $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 0,$$ such that the image of $A$ lies in the ...
12
votes
0
answers
380
views
Record analytic rank for an elliptic curve?
What is the current record (and reference) for the highest analytic rank of an elliptic curve over $\mathbb{Q}$?
The highest algebraic rank is the Elkies curve with rank at least 28, but I cannot ...
12
votes
0
answers
281
views
Statistics for rational points on curves of genus $g$ over $\mathbb{F}_q$, $g\gg q$
Consider the distribution of the number of $\mathbb{F}_q$ points as I range over smooth projective curves of genus $g$ (defined over $\mathbb{F}_q$). If $q\gg g,$ the Hasse-Weil bounds give me a lot ...
12
votes
0
answers
1k
views
Euler's totient function and Riemann hypothesis
I am looking for an upper-bound of the Euler's totient function $\varphi$ which would be equivalent to the Riemann hypothesis (RH). There is the following Nicolas' criterion about primorial numbers $...
12
votes
0
answers
309
views
Writing integers as a product of as few elements of $\{\frac21, \frac32, \frac43, \frac54, \ldots\}$ as possible
Is the number of elements we need to construct $x$ equal to $\log_2(x) + O(1)$?
This question is inspired by question 2 of the 2018 European Girls' Mathematical Olympiad. I previously posted it on ...