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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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, ...
Bernardo Recamán Santos's user avatar
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],...
Q_p's user avatar
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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?
user139770's user avatar
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 ...
Mark Fischler's user avatar
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_\...
Daniel Litt's user avatar
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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=...
Wolfgang's user avatar
  • 13.2k
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 $$ 1+3^a = 3^b+5^c. $$
hookah's user avatar
  • 1,096
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 ...
Bernardo Recamán Santos's user avatar
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 ...
Y. Zhao's user avatar
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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 ...
Q_p's user avatar
  • 824
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\...
Shimrod's user avatar
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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?
going_full_proper's user avatar
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 ...
Q_p's user avatar
  • 824
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 ...
ckamath's user avatar
  • 283
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 ...
Sylvain JULIEN's user avatar
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 ...
joro's user avatar
  • 24.2k
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?
Dmitry Krachun's user avatar
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 $\eta(k):=\sum_jk_j$. ...
T. Amdeberhan's user avatar
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 ...
Gerhard Paseman's user avatar
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 ...
A. Walker's user avatar
  • 161
-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$\colon$ $$ f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0. $$ ...
Pierre's user avatar
  • 563
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 ...
Maurizio Moreschi's user avatar
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 $E$ ...
Hanson S.'s user avatar
-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 ...
Jeffrey Ruppel's user avatar
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^{\...
asd's user avatar
  • 189
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 "...
joro's user avatar
  • 24.2k
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 $$ \...
user119197's user avatar
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 $$\{\sum_{j=1}^m j^2: m\in\mathbb{N}\} ?$$
Dominic van der Zypen's user avatar
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 ...
W-t-P's user avatar
  • 550
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,...
Zhi-Wei Sun's user avatar
  • 14.4k
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 ...
E.R's user avatar
  • 21
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 $Q$...
Fedor Petrov's user avatar
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 $\...
user138266's user avatar
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 ...
user138264's user avatar
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? ...
Tim Campion's user avatar
  • 60.6k
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 ...
Mike Battaglia's user avatar
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 &...
Seva's user avatar
  • 22.8k
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 ...
David Loeffler's user avatar
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, ...
Matthew Lim's user avatar
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 ...
Bernardo Recamán Santos's user avatar
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^...
Shimrod's user avatar
  • 2,335
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 ...
Nilotpal Kanti Sinha's user avatar
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,...
T. Amdeberhan's user avatar
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 ...
Yuri Bilu's user avatar
  • 1,130
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 ...
Yuri Bilu's user avatar
  • 1,130
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 \...
user36371's user avatar
  • 101
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 ...
macgucci's user avatar
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 ...
T. Amdeberhan's user avatar

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