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
15,955
questions
17
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2
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Is every odd positive integer of the form $P_{n+m}-P_n-P_m$?
I am looking for a comment, reference, remark, or proof of three conjectures as follows:
Conjecture 1: Let $x$ be an odd positive integer. Then there exist two integers $n, m \ge 2$ so that $$x=P_{n+...
CommunityBot
- 1
13
votes
0
answers
453
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The trace formula over function fields
There are many examples in number theory where an "arithmetic" problem (i.e. for number fields) has an easier analogue for function fields over finite fields. This is also true for questions ...
- 99.2k
9
votes
2
answers
326
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Algebraic power series over $\mathbb{F}_2$ as roots of polynomials of special form
Let $F = \mathbb{F}_2$ be the field with two elements. I will denote the rings of polynomials and formal
power series over $F$ as $F[t]$ and $F[[t]]$ respectively. Suppose that $x \in F[[t]]$ is ...
- 2,559
0
votes
1
answer
140
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Solutions to linear equations from recurrence sequences with no repeated roots
Let $U=(u_n)_{n=0}^{\infty}\subseteq\mathbb{C}$ be a sequence enumerated by a linear homogeneous recurrence relation with constant coefficients, i.e., there is some $d\geq 1$ and $a_1,\ldots,a_d\in\...
- 30k
1
vote
1
answer
150
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Simultaneous rational approximation to transcendental and algebraically independent numbers
I’m interested in the problem of simultaneous rational approximation to $k\geq 2$ numbers $\alpha_1,…,\alpha_k$ in the generic case where the $\alpha_j$ are transcendental and algebraically ...
- 103k
1
vote
1
answer
120
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On the regularity of integer solutions of a simultaneous equation with consecutive prime coefficients
Let $p_1$ through $p_6$ be consecutive primes in ascending order, and consider the simultaneous equation $$p_1x+p_2y=p_3\\p_4x+p_5y=p_6$$
Motivating Question: For what $p_1$ does the system provide ...
- 803
6
votes
2
answers
1k
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$\pi((n+1)^2)-\pi(n^2) \le \pi(n)$ for all $n \ge 370$?
There are some conjectures of the form: There always exist at least $X$ prime numbers between $A$ and $B$. Examples:
Bertrand's postulate: for every $n>1$ there is always at least one prime $p$ ...
- 43.3k
2
votes
1
answer
513
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Points of infinite level modular curve
Let us consider the anticanonical adic tower of modular curves which is described in Peter Scholze's paper "On torsion in the cohomology of locally symmetric varieties", and let us call $\mathcal{X}_{\...
- 5,550
4
votes
1
answer
411
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Is there some numerical evidence that $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ for any large enough $ x $?
Disclaimer : the following reasoning is a physicist's one and as such may not be suitable for this website. Still it may give rise to potentially interesting insights so I ask it anyway.
I stumbled ...
- 1,785
3
votes
2
answers
518
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Coefficients in Hirzebruch polynomial and divisibility of Bernoulli numbers: reference request
I seek a reference for the fact that "coefficients of the Hirzebruch $L$-polynomial have odd denominators". The coefficients are
$$\frac{2^{2k}(2^{2k-1}-1)B_k}{(2k)!}$$ where $B_k$ is the Bernoulli ...
CommunityBot
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3
votes
0
answers
203
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Cancellation in this exponential sum?
I would like to know whether it is possible to obtain cancellation in the sum
$$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$
where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.
- 12.7k
4
votes
0
answers
98
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Rank of binary matrix related to the number of positive squarefree integers less than $n$
I posted this question at the Mathematics SE, but received no response there so I am posting it here.
The following fact is stated in the comments-section of sequence A013928 in the OEIS.
Let $C$ ...
- 1,711
2
votes
0
answers
124
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Number of partitions of $\{1,2,\ldots,n\}$ whose blocks are arithmetic progressions of length $t$ or more
This question was inspired by the earlier question here,where no lower bound on arithmetic progression size was given. In particular, $t\geq 3,$ is assumed here.
The set $\{1,\ldots,n\}$ has $2^n$ ...
- 10.1k
20
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 ...
CommunityBot
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10
votes
1
answer
258
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The partial preorder on $\mathbb N$ generated by the finite axioms of choice
Let $\mathsf C_n$ denotes the statement:
for any family $\mathcal F$ of $n$-element sets there exists a choice function (i.e., a function $f:\mathcal F\to\bigcup\mathcal F$ such that $f(F)\in F$ for ...
- 40.9k
24
votes
2
answers
2k
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History of Geometric Analogies in Number Theory
My question, put simply, is: When did mathematicians/number theorists begin viewing questions in number theory through a geometric lens?
For example, was it before Grothendieck introduced schemes to ...
- 10.3k
5
votes
4
answers
761
views
Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$
For a fixed positive integer $n$, the Diophantine equation
$$x^2 + y^2 + z^2 = n$$
was studied by Gauss in Disquisitiones Arithmeticae. As is known, this equation is intimately connected to the ...
- 25.4k
28
votes
2
answers
2k
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When did people start thinking of elliptic curves as groups?
I have been reading some old papers of Cassels and Selmer from around 1950, and they talk about generators of rational solutions to elliptic curves, in the sense of Mordell–Weil, but do not appear to ...
- 312
31
votes
3
answers
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What is the Katz-Sarnak philosophy?
It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some ...
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41
votes
2
answers
7k
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Current Status on Langlands Program
The Langlands Program was launched almost fifty years ago, and progress has been made gradually, much of it hard earned. Langlands himself wrote a survey on the functoriality conjecture in 1997, Where ...
- 22.7k
0
votes
1
answer
270
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The minimum of the maximum of a sequence of sinc functions
I apologise if this is trivial or well known to be impossible:
Can one find a finite set of integers $2\leq a_1<a_2<\ldots<a_m<\infty$
such that for the function defined as
$$
f_{a_1,\...
- 10.1k
43
votes
1
answer
3k
views
What is the status of Arthur's book?
Arthur's long-awaited book project is now published (The endoscopic classification of representations: orthogonal and symplectic groups). However, in the book he takes some things for granted:
The ...
- 5,709
4
votes
2
answers
440
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Is Riemann zeta function injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
Or more generally, are L-functions injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
- 99.2k
0
votes
4
answers
688
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On the real part of the Riemann zeta function inside the critical strip
Denote by $\zeta$ the Riemann zeta function. Does $\Re\zeta(s)$ ever vanish for $\frac{1}{2}<\Re(s)\leq 1$ ?
- 99.2k
12
votes
2
answers
986
views
counting points on unit sphere mod p
Let $f(n)$ be the number of points on the unit sphere $x^2 + y^2 + z^2 = 1\; \pmod n$ with $x,y,z \in \mathbb{Z}/n\mathbb{Z}$
This is sequence A087784 in the Online Encyclopedia of Integer ...
- 30.2k
4
votes
2
answers
750
views
asymptotic for restricted partitions
Let $m$ and $n$ be two positive integers and denote by $P(n,m)$ the number of partitions of $n$ into $m$ non-negative integers.
Is there an asymptotic formula for $P(n,m)$ ?? Any reference is welcome....
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7
votes
2
answers
971
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Consequences of a bound on possible counterexamples to Riemann hypothesis
The Riemann hypothesis has many strong consequences in number theory. The question is: would a bound on the number of zeros of Riemann zeta-function in the critical strip with real part not equal 1/2 ...
- 99.2k
26
votes
1
answer
4k
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Why did Euler consider the zeta function?
Many zeta functions and L-functions which are generalizations of the Riemann zeta function play very important roles in modern mathematics (Kummer criterion, class number formula, Weil conjecture, BSD ...
- 312
1
vote
0
answers
143
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Fejer-Jackson-like inequality with divisor sum
A question was recently asked about a generalization of the Fejer-Jackson inequality $$\sum_{k=1}^n \frac{\sin kx}{k}\gt 0 \quad \forall\: n\in\mathbb{Z}^+\: \text{and}\: 0\lt x\lt\pi$$
to ...
- 105
32
votes
1
answer
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Are there any integers which can't be written as a sum of two fourth powers minus a cube?
To be precise, I am asking:
Does there exist an integer $k$ such that there do not exist (possibly negative) integers $x,y,z$ satisfying $x^4+y^4=z^3+k$?
Heuristically the answer must be yes, in ...
- 14.5k
1
vote
1
answer
151
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Is $P_{2n} \ge 2P_n$ and $\pi(2x) \le 2\pi(x)$ inconsistent to the Prime k-tuple conjecture?
I posed a conjecture as follows that is a special case of Second Hardy–Littlewood conjecture:
For $n, x \ge 2$ be two integers then:
$$P_{2n} \ge 2P_n$$
and
$$\pi(2x) \le 2\pi(x)...
- 99.2k
-3
votes
1
answer
242
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Can this weakening of Polignac's conjecture be proven?
Let $ A $ be a set of odd primes such that between any two consecutive elements thereof there is at least one prime gap that occurs infinitely often, i.e. an even integer $ g $ such that the ...
- 14.5k
3
votes
0
answers
165
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Why is the smallest (fractional) absolute central moment of a Gaussian distribution almost at $\sqrt{3}/2$?
Let $X$ be a standard normal random variable. What $\alpha$ minimizes $E|X|^{\alpha}$?
Numerically, $\alpha$ turns out to be equal to $\sqrt{3}/2-\varepsilon$ where $\varepsilon$ is of the order $10^...
0
votes
1
answer
153
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Sergei numbers : even integers n being a prime gap at least n times
Let's introduce Sergei (for SElf-Referential Gaps Extensible to Infinity, and as a wink to a mathematician friend of mine of Russian descent whose given name is Serge and quite interested in number ...
CommunityBot
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2
votes
1
answer
220
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Sieving modulo non-prime residue classes
Let $n$ be a positive integer, and consider the set $\{1, \dots, n\}$. If we remove from this set all the numbers $a$ which satisfy
$$
a \equiv 0 \mod d
$$
for at least one divisor $d$ of $n$ (...
- 30.7k
7
votes
1
answer
412
views
On $\varphi(m)\varphi(n)\equiv0\pmod{m+n}$
Euler's totient function $\varphi$ is multiplicative, and it plays important roles in number theory.
QUESTION: Is it true that for each integer $m>6$ we have $\varphi(m)\varphi(n)\equiv0\pmod{m+n}$...
- 13k
3
votes
0
answers
186
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Factoring problem similar to $RSA$ structure that is possibly not $NP$ complete and not $coNP$ also?
Standard factoring problem $\Pi_1$ is 'Given integers $N$ and $M$ is there a factor $d\in[1,M]$ of $N$?'. This is in $NP$ since such a factor is the witness and in $coNP$ since one can check all the ...
- 13.7k
14
votes
1
answer
357
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How are MTCs permuted by the Galois action on the little disk operad?
There is a well-studied action of $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$ on (some version of) the $E_2$ operad; see for example this MO question.
Modular tensor categories are examples of $...
- 8,244
10
votes
0
answers
363
views
Local Langlands Correspondence for unramified principal series representations
Let $G$ be a connected, reductive group over a $p$-adic field $k$. Assume $G$ is quasi-split with Borel subgroup $B = TU$. Consider those irreducible admissible representations $\pi$ of $G(k)$ which ...
- 6,110
4
votes
2
answers
957
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Primes $p$ for which $2p-1$ is prime
It's a well-known open problem (Sophie-Germain primes) whether there are infinitely many primes $p$, $2p+1$. What about $p$, $2p-1$?
Seemingly it's also an open problem (see here and the linked ...
- 99.2k
9
votes
0
answers
337
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Is Videla's solution of Hilbert's tenth problem for rational functions over field of characteristic 2 wrong?
The paper in question.
Quick introduction to the problem: suppose that $F$ is a finite field of characteristic 2
(for purposes of this post $F = \mathbb{F}_2$ will suffice) and let $F[t]$ and $F(t)$ ...
- 543
6
votes
1
answer
579
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The Gauss Circle Problem asymptotic in dimension
The circle problem in $k$ dimensions: "For $n>0$, how many points $z\in \ \mathbb{Z}^k$ have $\|z\|^2\leq n$?"
For large $n$, the answer is $\approx n^{k/2}\cdot \operatorname{Vol}(B^k(0,1))+\...
- 178k
9
votes
1
answer
486
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Enquiry on a Diophantine problem
Let $x,y, z$ be relatively prime integers with $xyz \neq 0$. Suppose that
$$x^{m/n} + y^{m/n} = z^{m/n}$$
where $m,n$ are relatively prime integers with $mn \neq 0$.
Does it necessarily follow ...
- 4,745
1
vote
1
answer
462
views
Some divisibility constraints in Frobenius coin problem
Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$.
Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ excellent if linear form ...
CommunityBot
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8
votes
1
answer
375
views
Is $AA+A$ always at least as large as $A/A$?
Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A/A|$?
In the line above, $AA+A:=\{ab+c:a,b,c \in A \}$, while $A/A:=\{a/b:a,b \in A, b\neq 0 \}$ is the ratio set.
...
- 1,478
4
votes
1
answer
991
views
3-smooth number
Let $(u_n)_{n\in\mathbb N}$ be the sequence of $3$-smooth numbers (that is whole number that can be written as $2^a3^b$ with $a,b\in\mathbb N$) sorted by increasing order. I am looking for the ...
- 99.2k
4
votes
0
answers
250
views
What is known about stability of number theoretic statements for Beurling systems which are based on small perturbations of the ordinary primes
Beurling considered a sequence of reals $1<x_1<x_2<\cdots <$ as "primes" and then the ordered sequence of all products of these "primes" as "integers". Let us consider Beurling primes ...
- 24.2k
3
votes
1
answer
818
views
Sum of Legendre symbol over primes
Given some $X, Y\ge 1$ and some $d\le Y$ not a perfect square, is it possible to bound
$$\sum_{p\le X}\left(\frac{d}{p}\right)?$$
As long as $Y$ is not too large compared to $X$, I would expect that ...
- 99.2k
4
votes
0
answers
145
views
Number of nontrivial integral solutions to $f(x)=f(y)$
Let $f(x)\in\mathbb Z[x]$ be a nonconstant polynomial, and let $$g(x,y)=\frac{f(x)-f(y)}{x-y}\in\mathbb Z[x,y].$$ Let $N(B)$ denote the number of pairs of integers $(x_0,y_0)$ such that $1\le x_0,y_0\...
- 1,001
4
votes
1
answer
164
views
An inequality involving $k$-generalized Fibonacci numbers
I have worked on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, to solve completely the equation I have one complicated case and I proved ...
- 59.8k