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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Grothendieck-Teichmüller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures Grothendieck-Teichmüller conjecture: the morphism $$ G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T}) $$ is an isomorphism. Here $...
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The exponent of Ш of $y^2 = x^3 + px$, where $p$ is a Fermat prime

For $d$ a non-zero integer, let $E_d$ be the elliptic curve $$ E_d : y^2 = x^3+dx. $$ When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD, $$ \# Ш(E_p)...
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Constructing non-torsion rational points (over Q) on elliptic curves of rank > 1

Consider an elliptic curve $E$ defined over $\mathbb Q$. Assume that the rank of $E(\mathbb Q)$ is $\geq2$. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank $=$ algebraic ...
H A Helfgott's user avatar
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Grothendieck's Period Conjecture and the missing p-adic Hodge Theories

Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over $\mathbb Q$ to $\mathbb Q$-vectors spaces. They come with the ...
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On the first sequence without triple in arithmetic progression

In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence: It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
Sebastien Palcoux's user avatar
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Can a regular icosahedron contain a rational point on each face?

The title says it all: Is there a (regular) icosahedron containing a rational point on each of its faces? For other Platonic solids, the affirmative answer is easy. Indeed, regular tetrahedra, cubes, ...
Ilya Bogdanov's user avatar
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What does the theta divisor of a number field know about its arithmetic?

This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link). Let ...
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A generalisation of the equation $n = ab + ac + bc$

In a result I am currently studying (completely unrelated to number theory), I had to examine the solvability of the equation $n = ab+ac+bc$ where $n,a,b,c$ are positive integers $0 < a < b < ...
Jernej's user avatar
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Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?

The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
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Defining $\mathbb{Z}$ in $\mathbb{Q}$

It was proved by Poonen that $\mathbb{Z}$ is definable in the structure $(\mathbb{Q}, +, \cdot, 0, 1)$ using $\forall \exists$ formula. Koenigsmann has shown that $\mathbb{Z}$ is in fact definable by ...
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Peano Arithmetic and the Field of Rationals

In 1949 Julia Robinson showed the undecidability of the first order theory of the field of rationals by demonstrating that the set of natural numbers $\Bbb{N}$ is first order definable in $(\Bbb{Q}, +,...
Ali Enayat's user avatar
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Cubic function $\mathbb{Z}^2 \to \mathbb{Z}$ cannot be injective

It is easy to show, with an explicit construction, that a homogeneous cubic function $f: \mathbb{Z}^2 \to \mathbb{Z}$ is not injective. I am seeking a proof of the same result without the condition ...
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Is there any positive integer sequence $c_{n+1}=\frac{c_n(c_n+n+d)}n$?

In a recent answer Max Alekseyev provided two recurrences of the form mentioned in the title which stay integer for a long time. However, they eventually fail. QUESTION Is there any (added: ...
Ilya Bogdanov's user avatar
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A question related to the Hofstadter–Conway \$10000 sequence

The Hofstadter–Conway \$10000 sequence is defined by the nested recurrence relation $$c(n) = c(c(n-1)) + c(n-c(n-1))$$ with $c(1) = c(2) = 1$. This sequence is A004001 and it is well-known that this ...
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Is there an Ehrhart polynomial for Gaussian integers

Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various ...
David E Speyer's user avatar
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A modern perspective on the relationship between Drinfeld modules and shtukas

Shtukas were defined by Drinfeld as a generalization of Drinfeld modules. While the relationship between the definitions of Drinfeld modules and shtukas is not obvious, one does have a natural ...
Will Sawin's user avatar
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Linking formulas by Euler, Pólya, Nekrasov-Okounkov

Consider the formal product $$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$ (a) If $z=2$ then on the one hand we get Euler's $$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$ on the ...
T. Amdeberhan's user avatar
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Non-linear expanders?

Recall that a family of graphs (indexed by an infinite set, such as the primes, say) is called an expander family if there is a $\delta>0$ such that, on every graph in the family, the discrete ...
H A Helfgott's user avatar
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What are the possible singular fibers of an elliptic fibration over a higher dimensional base?

An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one. It is often required for the ...
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A sequence potentially consisting of only integers

I will first ask the question which can be stated very simply. Afterwards I will explain some motivation and give references to related sequences. Consider the sequence defined by $$b_n = \frac{(...
John Machacek's user avatar
28 votes
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696 views

Does this infinite primes snake-product converge?

This re-asks a question I posed on MSE: Q. Does this infinite product converge? $$ \frac{2}{3}\cdot\frac{7}{5}\cdot\frac{11}{13}\cdot\frac{19}{17}\cdot\frac{23}{29}\cdot\frac{37}{31} \cdot \cdots \...
Joseph O'Rourke's user avatar
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891 views

On certain representations of algebraic numbers in terms of trigonometric functions

Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ...
Vladimir Reshetnikov's user avatar
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Derivative of Class number of real quadratic fields

Let $\Delta$ be a fundamental quadratic discriminant, set $N = |\Delta|$, and define the Fekete polynomials $$ F_N(X) = \sum_{a=1}^N \Big(\frac{\Delta}a\Big) X^a. $$ Define $$ f_N(X) = \frac{F_N(X)}{...
Franz Lemmermeyer's user avatar
26 votes
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526 views

Elliptic analogue of primes of the form $x^2 + 1$

I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
Marty's user avatar
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Is the Flajolet-Martin constant irrational? Is it transcendental?

Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm. In the ...
Jeffrey Shallit's user avatar
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Which sets of roots of unity give a polynomial with nonnegative coefficients?

The question in brief:   When does a subset $S$ of the complex $n$th roots of unity have the property that $$\prod_{\alpha\, \in \,S} (z-\alpha)$$ gives a polynomial in $\mathbb R[z]$ with ...
Louis Deaett's user avatar
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Orders in number fields

Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$. Question: Let $p$ be an ...
Ramin's user avatar
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25 votes
0 answers
594 views

Galois representations attached to Shimura varieties - after a decade

In an answer to the question Tools for the Langlands Program?, Emerton, in his usual illuminating manner, remarks on the reciprocity aspect of Langlands Program: "...As to constructing Galois ...
Nimas's user avatar
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24 votes
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0's in 815915283247897734345611269596115894272000000000

Is 40 the largest number for which all the 0 digits in the decimal form of $n!$ come at the end? Motivation: My son considered learning all digits of 40! for my birthday. I told him that the best way ...
domotorp's user avatar
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24 votes
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How did Gauss find the units of the cubic field $\mathbb Q[n^{1/3}]$?

Recently I read the National Mathematics Magazine article "Bell - Gauss and the Early Development of Algebraic Numbers", which gives a good description of the genesis of Gauss's ideas ...
user2554's user avatar
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Exotic 4-spheres and the Tate-Shafarevich Group

The title is a talk given by Sir M. Atiyah in a conference with the following abstract: I will explain a deep analogy between 4-dimensional smooth geometry (Donaldson theory)...
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Nekrasov-Okounkov hook length formula

I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls the Nekrasov-...
Dianbin Bao's user avatar
24 votes
0 answers
803 views

Smooth proper schemes over Z with points everywhere locally

This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question. Question. Is there a smooth proper scheme $X\to\operatorname{...
Chandan Singh Dalawat's user avatar
23 votes
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12k views

Philosophy behind Zhang's 2022 preprint on the Landau–Siegel zero

Now that a week has passed since Zhang posted his preprint Discrete mean estimates and the Landau–Siegel zero on the arXiv, I'm wondering if someone can give a high-level overview of his strategy. In ...
Stopple's user avatar
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23 votes
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Is A276175 integer-only?

The terms of the sequence A276123, defined by $a_0=a_1=a_2=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)}{a_{n-3}}\;,$$ are all integers (it's easy to prove that for all $n\geq2$, $a_n=\frac{9-3(-1)^n}{2}...
uvdose's user avatar
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23 votes
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884 views

Base change for $\sqrt{2}.$

This is a direct follow-up to Conjecture on irrational algebraic numbers. Take the decimal expansion for $\sqrt{2},$ but now think of it as the base $11$ expansion of some number $\theta_{11}.$ Is ...
Igor Rivin's user avatar
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23 votes
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Eichler-Shimura over Totally Real Fields

By Eichler-Shimura over totally real fields I mean the following conjecture. Conjecture. Let $K$ be a totally real field. Let $f$ be a Hilbert eigenform with rational eigenvalues, of parallel weight $...
Siksek's user avatar
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22 votes
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Are Erdős polynomials irreducible?

Define the Erdős polynomial to be $f_n(x):= \sum \limits_{0 \leq i,j \leq n}^{}{x^{ij}}$ (the name is motivated by http://oeis.org/A027424). For example for $n=5$, the polynomial is given by $x^{25}+...
Mare's user avatar
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22 votes
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820 views

Whither Kronecker's Jugendtraum?

Kronecker's Jugendtraum (Hilbert's 12th problem) asks us to find for any number field $K$ an explicit collection of complex-valued functions whose explicitly described special values generate the ...
Nimas's user avatar
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22 votes
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Do we know how to determine the $2^{2020}$ decimal of $\sqrt{2}$?

In the case of $\dfrac{1}{7^{800}}$ it's easy, to find the $2^{2020}$ decimal, but what about the simplest of the irrational numbers. Question: Do we know how to determine the $2^{2020}$ decimal of $\...
Dattier's user avatar
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22 votes
0 answers
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Fake CM elliptic curves

Suppose one has an elliptic curve $E$ over $\mathbb{Q}$ with conductor $N < k^3$ for some (large) positive $k$, with the property that its Fourier coefficients satisfy $$ a_p=0, \; \mbox{ for all }...
Mike Bennett's user avatar
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22 votes
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Most "natural" proof of the existence of Hilbert class fields

Assume that you have proved the two inequalities of class field theory, and that you want to show that the Hilbert class field, i.e., the maximal unramified abelian extension, of a number field $K$ ...
Franz Lemmermeyer's user avatar
21 votes
0 answers
764 views

Class field theory and the class group

Let $k$ be a finite abelian extension of $\mathbb{Q}$. Class field theory states that $k$ corresponds to some open subgroup of finite index $U_k \subset \mathbb{A}_{\mathbb{Q}}^*/ \mathbb{Q}^*$ where $...
Daniel Loughran's user avatar
21 votes
0 answers
599 views

Bounding failures of the integral Hodge and Tate conjectures

It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what ...
Daniel Loughran's user avatar
21 votes
0 answers
2k views

K-theory and rings of integers

From the works of Borel and Quillen there is a connection between the $K$-theory of the ring of integers $\mathfrak{o}_K$ in a number field $K$ and the arithmetic of the number field. In fact, it is ...
Daniel Larsson's user avatar
21 votes
0 answers
891 views

Deciding whether a given power series is modular or not

The degree 3 modular equation for the Jacobi modular invariant $$ \lambda(q)=\biggl(\frac{\sum_{n\in\mathbb Z}q^{(n+1/2)^2}}{\sum_{n\in\mathbb Z}q^{n^2}}\biggr)^4 $$ is given by $$ (\alpha^2+\beta^2+6\...
Wadim Zudilin's user avatar
20 votes
0 answers
476 views

Is every positive integer the rank of an elliptic curve over some number field?

For every positive integer $n$, is there some number field $K$ and elliptic curve $E/K$ such that $E(K)$ has rank $n$? It's easy to show that the set of such $n$ is unbounded. But can one show that ...
David Corwin's user avatar
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20 votes
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954 views

Finiteness of etale cohomology for arithmetic schemes

By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$. Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite ...
Daniel Loughran's user avatar
20 votes
0 answers
877 views

Looking for an effective irrationality measure of $\pi$

Most standard summaries of the literature on irrationality measure simply say, e.g., that $$ \left| \pi - \frac{p}{q}\right| > \frac{1}{q^{7.6063}} $$ for all sufficiently large $q$, without giving ...
Micah's user avatar
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20 votes
0 answers
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Could unramified Galois groups satisfy a version of property tau?

This is an experiment: there is a question I want to mention in an article I'm writing, and I am not sure it's a sensible question, so I will ask it here first, in the hopes that if it's insensible ...
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