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
2,239
questions
65
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Can you solve the listed smallest open Diophantine equations?
In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and ...
- 4,905
49
votes
2
answers
10k
views
Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?
Mathematica knows that:
$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$
The von Mangoldt function should then be:
$$\Lambda(n)=...
33
votes
3
answers
11k
views
About Goldbach's conjecture
let's consider a composite natural number $n$ greater or equal to $4$. Goldbach's conjecture is equivalent to the following statement: "there is at least one natural number $r$ such as $(n-r)$ ...
- 6,954
106
votes
6
answers
18k
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How small can a sum of a few roots of unity be?
Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...
- 106k
9
votes
0
answers
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Is the conjecture A+B=C following correct?
Is the conjecture on A+B=C following correct ?
Conjecture: Let $A, B, C$ be three positive integer numbers such that $A+B=C$ with $\gcd(A, B, C) = 1$. By Fundamental theorem of arithmetic we write:
$...
- 1,076
168
votes
3
answers
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Estimating the size of solutions of a diophantine equation
A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ?
(I do not know any such numbers).
B. Suppose that $\frac{a}{b+c} + \...
- 1,821
54
votes
6
answers
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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 ...
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49
votes
4
answers
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Is there an "elementary" proof of the infinitude of completely split primes?
Let $K$ be a Galois extension of the rationals with degree $n$. The Chebotarev Density Theorem guarantees that the rational primes that split completely in $K$ have density $1/n$ and thus there are ...
- 43.5k
42
votes
3
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Which integers can be expressed as a sum of three cubes in infinitely many ways?
For fixed $n \in \mathbb{N}$ consider integer solutions to
$$x^3+y^3+z^3=n \qquad (1) $$
If $n$ is a cube or twice a cube, identities exist.
Elkies suggests no other polynomial identities are known.
...
- 24.1k
13
votes
3
answers
1k
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Bound the error in estimating a relative totient function
Let $n=p_1^{e_1}\cdots p_k^{e_k}$ be an integer with $k$ prime factors. We know that the number of integers less than $n$ and coprime to it is
$$\Phi(n)=n-\sum_i\frac n{p_i}+\sum_{i \lt j}\frac n{...
- 29.9k
12
votes
0
answers
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Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?
Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, (n-r,n+...
- 6,954
114
votes
8
answers
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Zagier's one-sentence proof of a theorem of Fermat
Zagier has a very short proof (MR1041893, JSTOR) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an involution of the set $S= \lbrace (x,y,z) \...
- 6,014
44
votes
2
answers
7k
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What is known about the sum x^{n^2}/n?
It follows from a general theorem of Honda that the formal group with the logarithm
$$
x+x^{2^s}/2+x^{3^s}/3+x^{4^s}/4+\cdots
$$
has integer coefficients. I became interested in it because its $p$-...
- 17.3k
35
votes
3
answers
7k
views
Distinct numbers in multiplication table
Consider the multiplication table for the numbers $1,2,\dots, n$. How many different numbers are there? That is, how many different numbers of the form $ij$ with $1 \le i, j \le n$ are there?
I'm ...
- 2,741
26
votes
7
answers
6k
views
Asymptotic density of k-almost primes
Let $\pi_k(x)=|\{n\le x:n=p_1p_2\cdots p_k\}|$ be the counting function for the k-almost primes, generalizing $\pi(x)=\pi_1(x)$. A result of Landau is
$$\pi_k(x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\...
- 8,914
22
votes
3
answers
2k
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Hecke equidistribution
For a prime $p\equiv 1\pmod{4}$, we can write $p=a^2+b^2=N(a+bi)$. Therefore
$$
a+bi=p^{1/2}e^{i\varphi}
$$
where $\varphi\in [0,2\pi]$. I know that Hecke proved that $\varphi$ is equidistributed. I ...
- 2,458
18
votes
3
answers
4k
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sum of squares in ring of integers
Lagrange proved that every positive integer is a sum of 4 squares.
Are there general results like this for rings of integers of number fields? Is this class field theory?
Explicitly, suppose a ...
user2529
55
votes
3
answers
5k
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Number of elements in the set $\{1,\cdots,n\}\cdot\{1,\cdots,n\}$
Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?
- 867
39
votes
4
answers
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Which number fields are monogenic? and related questions
A number field $K$ is said to be monogenic when $\mathcal{O}_K=\mathbb{Z}[\alpha]$ for some $\alpha\in\mathcal{O}_K$. What is currently known about which $K$ are monogenic? Which are not? From Marcus'...
- 6,702
524
votes
3
answers
55k
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Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?
Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?
- 5,283
59
votes
5
answers
11k
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Is the Riemann Hypothesis equivalent to a $\Pi_1$ sentence?
1) Can the Riemann Hypothesis (RH) be expressed as a $\Pi_1$ sentence?
More formally,
2) Is there a $\Pi_1$ sentence which is provably equivalent to RH in PA?
Update (July 2010):
So we have two ...
- 5,264
49
votes
4
answers
4k
views
What fraction of the integer lattice can be seen from the origin?
Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$.
Say that a point $(x,y)$ of $Q$ is visible from the origin if the
segment from $(0,0)$ to $(x,y) \in Q$ passes ...
- 149k
45
votes
1
answer
5k
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Is it possible to show that $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges?
Let $\mu(n)$ denote the Mobius function with the well-known Dirichlet series representation
$$
\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}.
$$
Basic theorems about Dirichlet series ...
- 19.9k
22
votes
1
answer
2k
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The abc-conjecture as an inequality for inner-products?
The abc-conjecture is:
For every $\epsilon > 0$ there exists $K_{\epsilon}$ such that for all natural numbers $a \neq b$ we have:
$$ \frac{a+b}{\gcd(a,b)}\,\ <\,\ K_{\epsilon}\cdot \text{rad}\...
user6671
13
votes
2
answers
1k
views
distribution of coprime integers
Let $0 < a < 1$ be fixed, and integer $n$ tends to infinity. It is not hard to show that the number of integers $k$ coprime to $n$ such that $1\leq k\leq an$ asymtotically equals $(a+o(1))\...
- 99k
137
votes
2
answers
54k
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Consequences resulting from Yitang Zhang's latest claimed results on Landau-Siegel zeros
Very recently, Yitang Zhang just gave a (virtual) talk about his work on Landau-Siegel zeros at Shandong University on the 5th of November's morning in China. He will also give a talk on 8th November ...
- 1,503
291
votes
8
answers
140k
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Philosophy behind Mochizuki's work on the ABC conjecture
Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
77
votes
5
answers
18k
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Inaccessible cardinals and Andrew Wiles's proof
In a recent issue of New Scientist (16 Aug 2010), I was surprised to read that a part of Wiles' proof of Taniyama-Shimura conjecture relies on inaccessible cardinals.
Here's the article
Richard Elwes,...
- 1,031
58
votes
9
answers
15k
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Learning Class Field Theory: Local or Global First?
I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about ...
- 14.9k
58
votes
6
answers
6k
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Has decidability got something to do with primes?
Note: I have modified the question to make it clearer and more relevant. That makes some of references to the old version no longer hold. I hope the victims won't be furious over this.
Motivation:
...
- 2,734
48
votes
4
answers
4k
views
Are there primes of every Hamming weight?
Are there primes of every Hamming weight? That is, for every integer $n \in \mathbb{Z}_{>0}$ does there exist a prime which is the sum of $n$ distinct powers of $2$?
In this case, the Hamming ...
- 583
47
votes
1
answer
3k
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improving known bounds for Pierce expansions; cash prize
Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have ...
- 2,908
46
votes
5
answers
5k
views
The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$
I was asked the following question by a colleague and was embarrassed not to know the answer.
Let $f(x), g(x) \in \mathbb{Z}[x]$ with no root in common. Let $I = (f(x),g(x))\cap \mathbb{Z}$, that is, ...
- 30.1k
45
votes
7
answers
10k
views
Are some numbers more irrational than others?
Some irrational numbers are transcendental, which makes them in some sense "more irrational" than algebraic numbers. There are also numbers, such as the golden ratio $\varphi$, which are poorly ...
- 1,803
37
votes
8
answers
11k
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How does one motivate the analytic continuation of the Riemann zeta function?
I saw the functional equation and its proof for the Riemann zeta function many times, but usually the books start with, e.g. tricky change of variable of Gamma function or other seemingly unmotivated ...
- 3,738
34
votes
3
answers
4k
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Is there a ring of integers except for Z, such that every extension of it is ramified?
This is probably common knowledge, alas I have to confess my ignorance.
In simpler more abstract language, does $\mathcal{O}_K$ being simply connected (having trivial etale $\pi_1$) imply $\mathcal{O}...
- 6,148
28
votes
1
answer
1k
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Is the Golomb countable connected space topologically rigid?
The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $...
- 40.2k
28
votes
4
answers
5k
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distance between powers of 2 and powers of 3
I expect this is a classical question, so feel free to point me to classical answers: what is the fastest-growing function $f(t)$ for which we know that
$$
|2^t - 3^{t'}| \ge f(\min(t,t')) \;?
$$
In ...
- 413
26
votes
5
answers
2k
views
Are the 'semi' trivial zeros of $\zeta(s) \pm \zeta(1-s)$ all on the critical line?
The proof that $\Gamma(z)\pm \Gamma(1-z)$ only has zeros for $z \in \mathbb{R}$ or $z= \frac12 +i \mathbb{R}$ has been given here:
Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real ...
- 4,109
22
votes
5
answers
7k
views
Rational points on a sphere in $\mathbb{R}^d$
Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers.
Q1.
Are the rational points dense on the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$, i.e. does $S$ ...
- 149k
21
votes
4
answers
3k
views
Is the Euler product formula always divergent for 0<Re(s)<1?
It is known that the Euler product formula converges for $\Re(s)>1$
(and there it represents the Riemann zeta function).
My question: Is the Euler product always divergent for
$0 < \Re(s) < ...
- 255
20
votes
6
answers
4k
views
Erik Westzynthius's cool upper bound argument: update?
Version 2 of this writeup is
available, and includes a newer and simple upper bound thanks to
MathOverflow 88777 as
well as indirect references to future writeups. Details of further work
...
- 12.9k
20
votes
2
answers
2k
views
On a result attributed to W. Ljunggren and T. Nagell
I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation
$$\frac{x^{n}-1}{x-1} = y^{2}$$
doesn't admit solutions in ...
- 8,672
17
votes
1
answer
2k
views
What is known about the relationship between Fermat's last theorem and Peano Arithmetic?
As far as I know, whether Fermat's Last Theorem is provable in Peano Arithmetic is an open problem. What is known about this problem?
In particular, what is known about the arithmetic systems $PA + \...
- 4,743
5
votes
3
answers
2k
views
Goldbach conjecture and other problems in additive combinatorics
The field is also known as additive number theory. I am interested in sums $z=x + y$ where $x \in S, y\in T$, and both $S, T$ are infinite sets of positive integers. For instance:
$S = T$ is the set ...
- 3,058
113
votes
22
answers
36k
views
What's the "best" proof of quadratic reciprocity?
For my purposes, you may want to interpret "best" as "clearest and easiest to understand for undergrads in a first number theory course," but don't feel too constrained.
77
votes
9
answers
24k
views
Irreducibility of polynomials in two variables
Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this paper (Brindza and Pintér, On the irreducibility of some polynomials in ...
- 30.2k
76
votes
12
answers
12k
views
Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
- 114k
58
votes
1
answer
14k
views
Is the Green-Tao theorem true for primes within a given arithmetic progression?
Ben Green and Terrence Tao proved that there are arbitrary length arithmetic progressions among the primes.
Now, consider an arithmetic progression with starting term $a$ and common difference $d$. ...
- 3,529
55
votes
4
answers
4k
views
An interesting integral expression for $\pi^n$?
I came on the following multiple integral while renormalizing elliptic multiple zeta values:
$$\int_0^1\cdots \int_0^1\int_1^\infty {{1}\over{t_n(t_{n-1}+t_n)\cdots (t_1+\cdots+t_n)}} dt_n\cdots dt_1.$...
- 937