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
15,907
questions
11
votes
1
answer
458
views
Perfect Runs of Consecutive Integers
A run of 2 or more consecutive integers is said to be perfect if the sum of its terms equals the sum of all the their proper divisors, adding common divisors as often as they occur. For example, 672, ...
2
votes
2
answers
395
views
Robin's inequality and the zeros of the Riemann zeta function
Robin showed that if $a\in(1/2, 1]$ is the supremum of the real parts of the zeros of the Riemann zeta function $\zeta(s)$, then $f(x)=\Omega_{\pm} (x^{-b})$, where $b$ is some number on $(a-1/2, 1/2],...
2
votes
0
answers
124
views
Lower and upper central series and derived series of the absolute Galois group
What are the lower and upper central series and derived series of the absolute Galois group, $Gal(\overline{\mathbb{Q}}/\mathbb{Q}$), and which towers of number fields do they correspond to?
2
votes
1
answer
197
views
Proving largest power of $(a^2-ab+b^2)$ that divides $a^p-b^p-(a-b)^p$ for odd prime $n$
It is obvious that for odd $n \in \Bbb N$, $a^n-b^n-(a-b)^n$ is divisible by $ab(a-b)$ (with $n=1$ being a special case in which $a^n-b^n-(a-b)^n$ is zero). This can be viewed as a fact about ...
24
votes
1
answer
875
views
Universal homotheties for elliptic curves
Let $K$ be a number field and $E_1, \cdots, E_n$ elliptic curves over $K$. Let $\ell$ be a prime. Then there exists an element $\sigma \in \text{Gal}(\overline{K}/K)$ such that $\sigma$ acts on $T_\...
24
votes
1
answer
2k
views
Why these surprising proportionalities of integrals involving odd zeta values?
Inspired by the well known $$\int_0^1\frac{\ln(1-x)\ln x}x\mathrm dx=\zeta(3)$$ and the integral given here (writing $\zeta_r:=\zeta(r)$ for easier reading)$$\int_0^1\frac{\ln^3(1-x)\ln x}x\mathrm dx=...
11
votes
1
answer
598
views
Diophantine equation $3^a+1=3^b+5^c$
This is not a research problem, but challenging enough that I've decided to post it in here:
Determine all triples $(a,b,c)$ of non-negative integers, satisfying
$$
1+3^a = 3^b+5^c.
$$
1
vote
1
answer
205
views
Runs of consecutive numbers that are not relatively prime to their digital sum
It is well known that there can be at most 20 consecutive integers (in base 10) that are divisible by their digital sum, so called Harshad or Niven numbers.
How long can a run of consecutive ...
9
votes
0
answers
279
views
Searching for hypergeometric motives that split
Motivation: It seems that the splitting of a hypergeometric motive is closely related to some highly non-trivial hypergeometric identities discovered by Ramanujan, Guillera et al. The splitting of ...
13
votes
1
answer
2k
views
Why shouldn't this prove the Prime Number Theorem?
Denote by $\mu$ the Mobius function. It is known that for every integer $k>1$, the number $\sum_{n=1}^{\infty} \frac{\mu(n)}{n^k}$ can be interpreted as the probability that a randomly chosen ...
16
votes
2
answers
1k
views
Examples of problems where considering "discrete analogues" has provided insight or led to a solution of the original problem
The Kakeya conjecture posits that any Kakeya set in $\mathbb{R}^n$ has dimension $n$.
A discrete (finitized?) version of this problem is the Finite Field Kakeya conjecture, which was proved by Dvir ...
1
vote
0
answers
130
views
Centralizers of Cartan subgroup III
Let $\mathcal O$ be an order in an imaginary quadratic field $K$. Let $n$ be a positive integer. The multiplicative group $(\mathcal O/n\mathcal O)^\times$ acts on the module $\mathcal O/n\mathcal O\...
4
votes
1
answer
339
views
Finite image but not crystalline
What is an example of a $p$-adic representation of the absolute Galois group of a $p$-adic field that has finite image on the inertia subgroup, but is not crystalline?
5
votes
1
answer
350
views
What was the first elementary proof that $\pi(x)=o(x)$?
Denote by $\pi(x)$ the number of primes $\leq x$. I'm interested in knowing who came up with the first elementary proof that $\pi(x)=o(x)$.
I know that Chebyshev demonstrated elementarily before ...
18
votes
1
answer
857
views
Lagrange four-squares theorem --- deterministic complexity
Lagrange's four-squares theorem states that every natural number can be represented as the sum of four integer squares. Rabin and Shallit gave a randomised algorithm that finds one of these solutions ...
1
vote
0
answers
164
views
Can Riemann's explicit formula be generalized to semi-primes?
Following Isometry group of an integer I wonder if one can define a "mock zeta function" $\zeta_{V}$ (where $V:=(\mathbb{Z}/2\mathbb{Z})^{2}$ stands for "Vierergruppe", the German word for the Klein ...
0
votes
0
answers
74
views
Commutative rings with many nilpotent elements and efficient computation
Let $K$ be commutative ring. Assume that for natural $n$
there are $n$ nilpotent elements $y_i \in K$ satisfying
$y_i^2=0, y_i y_j=y_j y_i \ne 0$ and $\prod_1^ny_i \neq 0$.
Is it possible to compute ...
7
votes
1
answer
536
views
All quadratic forms of given genus over $\mathbb{Z}$
Given a (ternary) quadratic form over $\mathbb{Z}$ how can I find all quadratic forms (up to equivalence over $\mathbb{Z}$) in the same genus?
6
votes
1
answer
432
views
A binary hook-length formula?
This is purely exploratory and inspired by curiosity.
Setup: For an integer $k>0$, let $k=\sum_{j\geq0}k_j2^j$ be its binary expansion and denote the sum of its digits by $\eta(k):=\sum_jk_j$. ...
1
vote
0
answers
56
views
On divisibility conditions implying local coprimality conditions
This question is inspired by Bernardo Recaman's question Strings of consecutive integers divisible by 1, 2, 3, ..., N on intervals of $n$ integers being divisible by the integers $1$ through $n$. The ...
16
votes
3
answers
2k
views
Tower of moduli spaces in Scholze's theory
My question is related to another one I read here in Overflow. I am reading Scholze's papers about moduli spaces of $p$-divisible groups and elliptic curves, and I am very interested in the formal ...
-1
votes
1
answer
215
views
Finite extension of $K[[X]]$ and the norm
Let $R \colon= K[[X]]$ be a formal power series ring over a field $K$. We consider a monic polynomial $f(T) \in R[T]$ as follows$\colon$
$$
f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0.
$$
...
2
votes
3
answers
318
views
Mahler measures of values of polynomials
Let $K\ne \mathbb{Q}$ be a number field, let $\alpha\in \mathcal{O}_K$ and let $f(X)\in \mathcal{O}_K[X]$. Denote the Mahler measure by $M$.
Is there any known result about the comparison of the ...
6
votes
1
answer
594
views
Endomorphism rings of ordinary elliptic curves
Let's say $p$ is a prime and $t\neq 0$ is a trace of Frobenius that occurs over $\mathbb{F}_p$. The discriminant of the Frobenius polynomial is
$\Delta:=t^2-4p.$
So we obtain $4p=t^2-\Delta.$ If $E$ ...
-2
votes
2
answers
193
views
on generating prime numbers [closed]
I apparently need to rephrase the question(s), so here goes:
I'm an amateur mathematician, and I have been working for quite some time on finding a more efficient way of factoring large semiprime ...
3
votes
0
answers
168
views
Estimating integral of product of terms $\cos(t\log p)$
I would like to prove the following proposition from A. Harper's paper "Sharp conditional upper bound for moments of the Riemann Zeta Function"
Proposition.
Let $T$ be large and let $n=p_1^{\...
3
votes
2
answers
790
views
Finding coefficient of multivariate polynomial
$f(x_1,x_2,\ldots x_n)$ is polynomial with integer coefficients.
$f$ is rather large to be computed explicitly, but an algorithm can
compute it efficiently at integers and complex number and "...
4
votes
0
answers
199
views
Fourier coeffients of Cantor measure
For $0<\theta<\frac{1}{2}$, denote by $\mu_\theta$ the uniform Cantor measure with dissection ratio $\theta$. It is not hard to show that the Fourier–Stieltjes transform of $\mu_\theta$ is
$$
\...
4
votes
1
answer
349
views
Squares in the set $\{\sum_{j=1}^m j^2: m\in\mathbb{N}\}$ [closed]
Are there infinitely many squares in the set $$\{\sum_{j=1}^m j^2: m\in\mathbb{N}\} ?$$
5
votes
1
answer
322
views
Reversing the CRT: Is $5$ tough?
Given odd primes $p\ne q$, by the CRT we can find an integer $x$ such that $x\equiv 2^{p-1}\pmod q$ and $x\equiv 2^{q-1}\pmod p$. Can this procedure be reversed?
For which integers $x$ there exist ...
4
votes
0
answers
171
views
On sums of minima and maxima
Let $h_1,\ldots,h_n$ be positive integers, and define
$$m(h_1,\ldots,h_n)=\sum_{r_1=0}^{h_1-1}\ldots\sum_{r_n=0}^{h_n-1}\min\left\{\frac{r_1}{h_1},\ldots,\frac{r_n}{h_n}\right\}$$
and
$$M(h_1,\ldots,...
2
votes
0
answers
93
views
The prime spectrume of integral-valued polynomial ring
Let $ D $ be an integral domain with quotiont field $K $ and let $Int (D) $be the set of all integral-valued polynomials on $D $, that is, $ Int (D):=\{f \in K[x]\mid f (D) \subseteq D\} $. The ...
7
votes
3
answers
665
views
Integer positive definite quadratic form as a sum of squares
Let $A$ be a symmetric $d\times d$ matrix with integer entries such that the quadratic form $Q(x)=\langle Ax,x\rangle, x\in \mathbb{R}^d$, is non-negative definite. For which $d$ does it imply that $Q$...
7
votes
0
answers
223
views
Automorphism group of poset of number fields
Consider the poset of number fields, partial order being defined by inclusion of fields. What is the group of order-preserving automorphisms of this poset? What if we take only Galois extensions of $\...
7
votes
0
answers
218
views
Space of algebraic closures of $\mathbb{Q}$
The ambiguity inherent in defining the absolute Galois group $G_\mathbb{Q}$ - that it is determined only up to inner automorphisms - arises from the fact that one has to choose an algebraic closure of ...
3
votes
1
answer
167
views
Complexity of generalizations of primality testing?
There is a known polynomial-time algorithm for testing whether an integer is prime. I'm wondering what is known about problems in this neighborhood:
Questions:
What happens over other number fields? ...
0
votes
1
answer
497
views
Order of magnitude of $\sum \frac{1}{\log^2{p}}$, or $\sum \frac{1}{\log^a{p}}$ for arbitrary power $a$ [closed]
In this MO question, it says that we have
$$ \sum_{p<n} \frac{1}{\log{p}} =\frac{n}{\log^2 n}+O\left(\frac{n\log\log n}{\log^3 n}\right).$$
where the sum is on all primes $p$, up to some max ...
10
votes
2
answers
1k
views
Reference request: Oldest number theory books with (unsolved) exercises?
Per the title, what are some of the oldest number theory books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the ...
2
votes
1
answer
184
views
A Vandermonde-type system
For a prime $p$ and $a_1,\dotsc,a_n\in\mathbb F_p^\times$, consider the system of equations
$$ \begin{cases}
\begin{align}
a_1 + \dotsb + a_n &= 0 \\
a_1x_1 + \dotsb + a_nx_n &...
7
votes
1
answer
449
views
Rigid versus log-rigid cohomology for semistable varieties
If $K$ is a p-adic field, with maximal unramified subfield $K_0$, and $X$ is a proper semi-stable $O_K$-scheme, then there's a canonical way to make the special fibre $X_k$ into a log-scheme; and ...
0
votes
0
answers
83
views
Generating the digits in a base system by repeated multiplication of a number
The first 15 terms of the sequence {a_i} = 2^i are 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768. All of the digits in base-10, i.e. {0, ...
8
votes
4
answers
578
views
Covering the primes with pairs of consecutive integers
Is it true that for every sufficiently large positive integer $n$, one can always find at most $k=\lfloor\pi(n)/2\rfloor$ integers, $a_1,a_2,a_3,a_3,\dots a_k$, between $1$ and $n$, such that each of ...
2
votes
0
answers
70
views
Centralizers of Cartan subgroups II
Let $K$ be an imaginary quadratic field and let $\mathcal O$ be its ring of integers. Suppose that $2$ is split in $\mathcal O$. Let $k$ be a positive integer. The multiplicative group $(\mathcal O/2^...
23
votes
1
answer
3k
views
Does the average primeness of natural numbers tend to zero?
This question was posted in MSE. It got many upvotes but no answer hence posting it in MO.
A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a value ...
7
votes
1
answer
395
views
Divisibility of sum of multinomials
Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
$$S(n,m,t)=\sum_{k_1+\cdots+k_n=m}\binom{m}{k_1,\dots,k_n}^t$$
where the sum runs over non-negative integers $k_1,\dots,...
19
votes
1
answer
1k
views
Explicit version of the Burgess theorem
Does there exist a totally explicit version of the Burgess theorem? Precisely, let $m$ be a positive integer, and let $\chi$ be a primitive character mod $m$. A special case (sufficient for my ...
13
votes
1
answer
740
views
Least quadratic residue under GRH: an explicit bound
Let $m$ be a positive integer and $\chi$ a primitive character mod $m$. Let $x$ be such that $\chi(p)\ne 1$ for all primes $p<x$. Assume GRH. How can one bound $x$ in terms of $m$ ? I do not need ...
3
votes
0
answers
213
views
Mod p reduction of geometrically irreducible polynomials
Let $f\in \mathbb Z[t,x]$ be a polynomial of positive degree that is irreducible over $\overline{\mathbb Q}[t,x]$. Is it true that for all but finitely many primes $p$ the reduced polynomial $f_p\in \...
1
vote
0
answers
178
views
An integral involving $1/\zeta(s)$ and the zeros of $\zeta(s)$
In my thesis, i stumbled across the following problem:
Define $$F(\sigma, T)=\frac{1}{T}\int_{-T}^{T} \frac{1}{|\zeta(\sigma+it)|^2} \mathrm{d}T$$ where $\zeta$ denotes the Riemann zeta function.
Is ...
2
votes
1
answer
143
views
From recursive polynomials to a $q$-series
Fix an integer $k$. Let $b_n(q)$ be the polynomial defined by the recursive equation
$$b_n(q)=\binom{n+k-1}k+(1+q^{n-1})b_{n-1}(q), \qquad n\geq1,$$
initializing with $b_0(q)=0$.
I run into the ...