# 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

2,948
questions with no upvoted or accepted answers

**19**

votes

**0**answers

452 views

### Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space

Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all
$x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...

**19**

votes

**0**answers

844 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\...

**19**

votes

**0**answers

2k views

### sums of digits of powers of integers

It is known (Senge and Straus, 1971, see also C.L.Stewart, 1980) that for every natural $a $, not a power of 10, and every natural $s$, there are only finitely many $k$ such that the sum of decimal ...

**18**

votes

**0**answers

230 views

### Geometry of algebraic curve determined by point counts over all number fields?

Let $C$ be a smooth irreducible projective curve of genus $g>1$ over a number field $K$. The Mordell conjecture (first proved by Faltings) says that for any finite field extension $L/K$ (say, ...

**18**

votes

**0**answers

895 views

### Number Theory and Gravity

Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands at IAS (1967, 1970), it seeks to relate Galois ...

**18**

votes

**0**answers

1k views

### $G$ a group, with $p$ a prime number, and $|G|=2^p-1$, is it abelian?

During my research I came across this question, I proposed it in the chat, but nobody could find a counterexample, so I allow myself to ask you :
$G$ a group, with $p$ a prime number, and $|G|=2^p-1$...

**18**

votes

**0**answers

624 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 ...

**18**

votes

**0**answers

473 views

### Why $\gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}$ likes to be large?

For a prime $p$, let $F_p$ denote the greatest common divisor of the orders modulo $p$ of all prime divisors of $p-1$:
$$ F_p = \gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}; $$
thus, for instance, $F_3=...

**18**

votes

**0**answers

598 views

### The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with $n=...

**18**

votes

**0**answers

999 views

### Does this variant of a theorem of Hasse (really due to Gauss) have an “elementary” proof?

BACKGROUND
Here are 3 theorems of varying difficulty. Let $M$ be the $Z/2$ subspace of $Z/2[[x]]$ spanned by $f^k$, with the $k>0$ and odd, and $f=x+x^9+x^{25}+x^{49}+\cdots$. For $g$ in $M$, let $...

**18**

votes

**0**answers

628 views

### Erdos-Kac for squarefree numbers

In its usual form, the Erdos-Kac Theorem states that if $f(n) : \mathbb{N} \rightarrow \mathbb{R}$ is a strongly additive function with $|f(p)| \le 1$ for all primes $p$, then
$$\frac{|\{n \le x : \...

**18**

votes

**1**answer

952 views

### A set of generators for $\bar{\mathbb{Q}}$

Two questions:
Does there exist a sequence $\alpha_1,\alpha_2,...$ of algebraic numbers with degrees $d_1,d_2,...$ s.t. for each $i$, $d_i|d_{i+1}$ and $\alpha_i= p_i(\alpha_{i+1})$ with $p_i$ a ...

**17**

votes

**0**answers

848 views

### Almost monochromatic point sets

There are many sort of equivalent theorems about monochromatic configurations in finite colorings, such as Van der Waerden, Hales-Jewett or Gallai's theorem, the latter of which states that in a ...

**17**

votes

**0**answers

482 views

### Tropicalization of perfectoid spaces and their tilts

Does tropicalization exist in the world of perfectoid spaces? Since it does for Huber's adic spaces, I thought it might for perfectoid spaces too, yet I can't find any explicit references so far.
...

**17**

votes

**0**answers

635 views

### Elements of finite fields with many powers of trace zero

Let $p$ be an odd prime number, $n>1$ be an integer, and $\mathrm{tr}$ be the trace map of the field extension $\mathrm{GF}(p^{2n})/\mathrm{GF}(p)$. For which pair $(p,n)$ does there exists $x\in\...

**17**

votes

**0**answers

1k views

### Does Mochizuki's proof of abc conjecture gives an upper bound for the quality of a triple?

The quality of a triple $(a,b,c)$ of coprime positive integers with $a + b = c$ is defined as
$$q(a,b,c) := \frac{\log(c)}{\log(\mathrm{rad}(abc))}.$$
Then
$$a+b = c = \mathrm{rad}(abc)^{q(a,b,c)}.$$...

**17**

votes

**0**answers

414 views

### Need explicit formula for certain “$q$-numbers” involving gcd's

The question is motivated by yet another possible approach to a combinatorial problem formulated previously in "Special" meanders. I'm not giving details of the connection as I believe the ...

**17**

votes

**0**answers

871 views

### Groups generated by 3 involutions

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$\tau_{r_1(m_1),r_2(m_2)}$ be the ...

**17**

votes

**0**answers

857 views

### Computation of low weight Siegel modular forms

We have these huge tables of elliptic curves, which were generated by computing modular forms of weight $2$ and level $\Gamma_0(N)$ as N increased.
For abelian surfaces over $\mathbb{Q}$ we have very ...

**16**

votes

**0**answers

462 views

### Shimura varieties and connected components

Let $G$ be a connected reductive algebraic group over $\mathbf{Q}$. I've seen two slightly different definitions in the literature of the Shimura variety of level $U$, for $U \subseteq G(\mathbf{A}_{\...

**16**

votes

**0**answers

756 views

### Why arithmetic Langlands?

In trying to understand the import of Akshay Venkatesh' most recent work I found myself wondering anew about that old gnawing mystery: why Langlands? Why should arithmetic of polynomial equations over ...

**16**

votes

**0**answers

511 views

### The number 1680 and Lagrange's four-square theorem

The number $1680$ has the factorization $2^4\times3\times5\times7$. Rather to my surprise, I found that this number has certain mysterious connection with Lagrange's four-square theorem.
QUESTION: ...

**16**

votes

**0**answers

568 views

### Infinite extensions such that every elliptic curve has finite rank

The comments to this answer seem to make the following claim.
Claim. Let $K$ be the maximal abelian extension of $\mathbf Q$ that is unramified away from $p$ (more generally, away from a finite set $...

**16**

votes

**0**answers

356 views

### Are there smooth and proper schemes over $\mathbb Z$ whose cohomology is not of Tate type

Is there an example of smooth and proper scheme $X \to \mathrm{Spec}(\mathbb Z)$, and an integer $i$ such that $H^i(X, \mathbb Q)$ is not a Hodge structure of Tate type?
Alternatively: such that $H^...

**16**

votes

**0**answers

664 views

### An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many primes $\equiv -1 \bmod n$

This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely ...

**16**

votes

**0**answers

432 views

### Are there any examples of hyperbolic curves over finite fields such that the action of frobenius on its prime-to-$p$ fundamental group is known?

Let $X$ smooth curve over a finite field $\mathbb{F}_q$ of type $(g,n)$ - that is, $X$ is an open subscheme of its genus $g$ compactification obtained by removing $n$ points.
Any such curve ...

**16**

votes

**0**answers

640 views

### Decidability of $x^3+y^3+z^3 = c$

I wondering if it is known whether the following problem is algorithmically decidable or undecidable by Turing machines: given an integer c, determine if there are integers $(x,y,z)$ such that $x^3+y^...

**16**

votes

**0**answers

765 views

### What is the expected dimension of the Zariski closure of the rational points on the moduli space of curves?

For each genus $g$, there are many curves of genus $g$ defined over $\mathbb Q$. How many? We might study this question by considering the rational points of the Deligne-Mumford moduli space of curves ...

**16**

votes

**0**answers

699 views

### Epsilon factors - a la Beilinson - What is it?

I understand, to some extent, Tate's thesis. Could somebody explain perhaps what are the epsilon factors in Beilinson's works, such as "$\epsilon$-factors for Gauss-Manin determinants", or "...

**16**

votes

**0**answers

409 views

### A Product Related to Unrestricted Partitions

Start with the product for unrestricted partitions:
$(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...$
Now replace some of the plus signs with minus signs and expand the product into a series. Is it ...

**16**

votes

**0**answers

703 views

### How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's
answer to the question Freeness of a Z[x] module depicted below? --
Choosing colors according to the highest power of 2 which ...

**16**

votes

**0**answers

698 views

### Lubin-Tate vs cohomological approach to local CFT

Local class field theory ("local CFT") can be developed in various ways, among them is a cohomological approach and an explicit approach due to Lubin and Tate (both can be found in Milne's CFT notes ...

**16**

votes

**0**answers

543 views

### Division fields of abelian varieties over function fields

Let $k$ be a finitely generated field (for example a finite field or a number field) and $K/k$ a finitely generated regular extension with $trdeg(K/k)=1$. Let $A/K$ be a principally polarized abelian ...

**16**

votes

**0**answers

1k views

### Can one compare integral structures on de Rham and crystalline cohomology?

Suppose $\mathfrak{X}$ is a smooth projective scheme of finite type over $\mathbb{Z}_p$, with generic fibre $X$. Then there are comparison theorems relating de Rham and crystalline cohomology,
$H^i_{\...

**16**

votes

**0**answers

653 views

### How many Hecke operators span the Hecke algebra?

This is a generalisation of my earlier question about generators for the level 1 Hecke algebra.
Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb{Z})$, and $k \ge 1$ an integer. ...

**16**

votes

**0**answers

873 views

### Polynomials with presumably positive coefficients

After seeing that some positivity problems get their solutions on MO,
I am quite enthusiastic of posing my (and not only) problem of positive flavour.
In order to state it, I have to introduce the ...

**16**

votes

**0**answers

454 views

### Cohomological characterization of CM curves

In his 1976 classical Annals paper on $p$-adic interpolation, N. Katz uses the fact that if $E_{/K}$ is an elliptic curve with complex multiplications in the quadratic field $F$, up to a suitable ...

**15**

votes

**0**answers

733 views

### What is the smallest unsolved diophantine equation?

If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq ...

**15**

votes

**0**answers

383 views

### Étale cohomology of varieties in positive characteristic via singular cohomology

Suppose $X$ is a smooth scheme, not necessarily projective, over $\mathbb{Z}[1/N]$ for some integer $N\neq 0$. I would like to understand the cohomology groups $H^i_{ét}(X_{\overline{\mathbb{F}}_p}, \...

**15**

votes

**0**answers

511 views

### Does every integer $n>1$ have the form $a^2+b^2+3^c+5^d$ with $a,b,c,d$ nonnegative integers?

Lagrange's four-square theorem states that every nonnegative integer is the sum of four squares. I have tried to replace two of the four squares by two powers. This leads to my following question: ...

**15**

votes

**0**answers

328 views

### Complexity of a Fibonacci numbers discrete log variation

In my work I encountered the following
FIBMOD PROBLEM:
Given $k,m$ in binary, decide if there exists $n$ such that
$\, F_n = k \,$ (mod $m$). Here $F_n$ is a Fibonacci number.
This is a variation ...

**15**

votes

**0**answers

420 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 ...

**15**

votes

**0**answers

361 views

### Relation between the Galois group of a polynomial and the symmetry of its roots

Let $f(x)$ be a polynomial with integer coefficients and non-zero discriminant (so each root of $f$ has multiplicity one). Consider the projective roots $\theta_1, \cdots, \theta_n$ of $f$, as ...

**15**

votes

**0**answers

614 views

### Is this set computable?

The starting point of this question is the observation that the smallest positive integers $a,b,c$ satisfying
$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$$
are absurdly high. This leads to ...

**15**

votes

**0**answers

429 views

### Word complexity of primes mod 4

For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...

**15**

votes

**0**answers

921 views

### Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac 1N]...

**15**

votes

**0**answers

350 views

### What is the simplest known arithmetic definition of exponentiation?

For $n$ a natural number, let $E_n$ denote the set of bounded arithmetic formulas consisting of n alternating blocks of bounded quantifiers starting with an existential quantifier, followed by a ...

**15**

votes

**0**answers

494 views

### For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?

An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...

**15**

votes

**0**answers

486 views

### What can be the dimension of a pointless smooth proper Z-scheme?

What is the smallest dimension $d$ such that there is a smooth proper morphism $X \to \operatorname{Spec} \mathbb Z$ of relative dimension $d$, with $X$ nonempty, without a section?
Of course, there ...

**15**

votes

**0**answers

611 views

### Bloch-Kato conjecture and Wiles' numerical criterion

I already asked this question some days ago on https://math.stackexchange.com/questions/158747/bloch-kato-conjecture-and-wiles-numerical-criterion but didn't receive any response.
In the ...