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
15,952
questions
1
vote
0
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49
views
Effective Erdős–Kac theorem
I have some number $N$ and some integer $k>0$. I want to know what fraction of numbers up to $N$ have more than $k$ prime factors. (In my application, with repetition, but the $\omega$ version is ...
- 8,994
1
vote
0
answers
30
views
automorphisms and mellin transforms
If a real analytic function $f$ is involutive i.e. $f(f(x))=x$ and its Mellin transform can be taken on a section of the real axis, and is analytic for $x>0$, in certain cases can this imply that $\...
- 101
6
votes
2
answers
216
views
A question on Euler's totient function
With reference to the Euler's totient function $\phi(\cdot)$, given any $n \in \mathbb{Z}^+$, it's quite straightforward to find $\phi(n)$.
In contrast, given $n \in \mathbb{Z}^+$, even though there ...
- 161
2
votes
0
answers
41
views
bijection from vectors with non-negative integer integer entries to integers
I have the following question. Given a natural number $N$ we construct a set $K$ of vectors of infinite length with non-negative integer entries with a given sum $N$. For example, for $N=3$ the set $K$...
- 61
5
votes
1
answer
155
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What integers can be represented as $\textrm{lcm}(a, b) + \textrm{lcm}(b, c) + \textrm{lcm}(c, a)$?
Find all integers $n$ that can be written as $n = \textrm{lcm}(a, b) + \textrm{lcm}(b, c) + \textrm{lcm}(c, a)$ where $a$, $b$, $c$ are all integers.
- 51
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votes
0
answers
63
views
Which algebraic structure characterizes the set of non-trivial qudratic residues in a finite field?
I understand this question may be too naive to ask, but I am unable to figure it out.
Suppose, $\mathbb{QR^*}$ denotes the set of all quadratic residues in a finite field except the identity element $...
2
votes
2
answers
205
views
Negated Fibonacci and the floor function
Let $F_n$ be A000045 (i.e., Fibonacci numbers). Here
$$
F_n = F_{n-1} + F_{n-2}, \\
F_0 = 0, F_1 = 1, \\
F_{-n} = (-1)^{n-1}F_n
$$
I conjecture that
$$
F_{-n} = \left\lfloor\frac{n+1}{2}\right\rfloor ...
- 3,464
3
votes
0
answers
113
views
Third roots of unity and norm element
Let $K = \mathbb{Q}(\zeta_3)$ where $\zeta_3$ is a third root of unity, let $F = \mathbb{Q}(\sqrt[3]{\ell})$ where $\ell \equiv 1 \pmod{9}$ is a prime and set $L$ to be the Galois closure of $F$, i.e.,...
- 1,191
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0
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54
views
Relation between exponents to the different bases over $\mathbb{Z}^\times_p$?
This question is similar to this question,
Relation between elements with fixed exponent over different $\mathbb{Z}^\times_p$
For each prime $p$ that has a primitive root $3$ and for all $a\in\mathbb{...
-1
votes
0
answers
96
views
Is it possible to have square-free order(s) in $\mathbb{Z}^\times_N$?
Suppose, $N=p\cdot q$ is the product of two safe primes $p=2p'+1$ and $q=2q'+1$ for some odd primes $p'$ and $q'$.
Let, $p_0,p_1,\ldots,p_m\ll p',q'$ be a few odd primes chosen uniformly at random ...
2
votes
1
answer
569
views
Separating Gamma in two independent functions
I've encountered a problem in my PhD. I would greatly appreciate any suggestions, tips, or comments you might have. The problem is
Let $\Gamma(s,x)$ be the incomplete gamma function. Given integers $n ...
- 31
2
votes
1
answer
144
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Small solutions of $x^2-a^3 y^2=\pm 1$
We are interested in small integer solutions to the Pell equation:
$$x^2-a^3 y^2=\pm 1 \qquad (1)$$
Where in $\pm 1$ you can chose either sign.
$(x^2,a^3 y^2)$ are consecutive powerful numbers.
$abc$ ...
- 24.2k
3
votes
0
answers
142
views
Cardinality of the set $\#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}$
Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$.
I am interested in upper bound for
$$
M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}
$$
where $N$ ...
- 3,595
4
votes
1
answer
174
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heights of ideal classes and reduction theory for Bhargava cubes
Suppose $K$ is a quadratic imaginary field with discriminant $D$; let $S$ denote the ring of integers in $K$. For a fractional $S$-ideal $J$, define the height of $J$, denoted $H(J)$, to be the ...
- 123
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votes
0
answers
172
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On a A057985 without recursion
Let $a(n)$ be A057985 (i.e., start with $0$ and repeatedly substitute: $0\to01, 1\to12, 2\to0$).
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). Here
$$
\...
- 3,464
0
votes
0
answers
80
views
Sum of odd reciprocals [duplicate]
Can (n-1)/n be expressed as sum of random "odd" distinct reciprocals ever? here 'n' is also an odd.
20
votes
0
answers
455
views
Low-level proof of identity related to Weierstrass P-function
A theorem which can be extracted from Theorem V.1.1 of Silverman's "advanced topics in the theory of elliptic curves" is the following. Here $\mathbb{Q}(u)$ denotes rational functions in a ...
- 40.8k
2
votes
0
answers
116
views
Tensor product of finite extensions of $\mathbb{Q}_p$
Consider the tensor product of finite extensions of a field $F$ of characteristic zero. (I am interested in the case $F=\mathbb{Q}_p$.)
$(1)$ If $M$ is a finite Galois extension of $F$ with Galois ...
- 424
6
votes
1
answer
318
views
Integrality of a quotient of Fermat numbers
I try to prove that for every positive integers $m\ge n$, the following product is an integer:
$$\prod_{k=0}^{n-1}\frac{2^{2^m}-2^{2^k}}{2^{2^n}-2^{2^k}}.$$
But no luck.
- 3,739
2
votes
0
answers
209
views
Representability of moduli problem of elliptic curves with complex multiplication
I'd like to know whether the moduli problem for elliptic curves with complex multiplication by a fixed imaginary quadratic number field $K$ (and with suitable level structure to be picked) is ...
- 79
2
votes
1
answer
147
views
Subset of $\mathbb N$ missing at least a class modulo each prime
One of my students asked me the following question. It seemed easy to answer but in fact, I am stucK.
The question: does there exist an infinite subset $S$ of $\mathbb N$ such there exists a positive ...
- 3,739
18
votes
0
answers
596
views
Consecutive integers of the form $2^a 3^b 5^c$
Let $\mathcal{N}$ denote the set of all products of (powers of) $2,3$ and $5$:
$$ \mathcal{N} = \{ 2^a 3^b 5^c \ : \ a,b,c \geq 0 \} \subset \mathbb{N}.$$
We use the elements of $\mathcal{N}$ to ...
- 1,762
1
vote
0
answers
69
views
Percolative process distribution not equivalent to coupon collector problem distribution
I have a process where; given a $n\times 1$ matrix initially empty, an element is inserted in it at a random position, with the possibility of repeating the insertion at a filled cell. Then, after a ...
- 61
1
vote
0
answers
62
views
Congruence obstructions for three consecutive powerful numbers
Powerful number is integer $m$ such that if $p \mid m$ then $p^2 \mid m$.
Powerful numbers can be represented in the form $m=u^2 v^3$.
Erdos conjectured that three consecutive powerful numbers don't
...
- 24.2k
3
votes
2
answers
207
views
Unique "clique" of differences in $\mathbb{Z}/m\mathbb{Z}$
Are there absolute constants $0 < \epsilon < 1$ and $N \in \mathbb{N}$ such that the following holds: For every $m \in \mathbb{N}$ and every $A \subseteq \mathbb{Z}/m\mathbb{Z}$ with $\frac{\...
- 73
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votes
1
answer
98
views
Could I possibly exploit distinct odd primes raised to 6 to solve Exact Three Cover, when reducing it in Subset Sum?
I'm solving Exact 3 Cover, given a list with no duplicates $S$ of $3m$ whole numbers and a collection $C$ of subsets of $S$, each containing exactly three elements. The goal is to decide if there are $...
- 101
-3
votes
0
answers
63
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On Ideals over The Ring of Integers of a Number Field [closed]
Alrighty I really just need to run this by someone to make sure I'm not saying something crazy.
First Consider $K$ a finite extension of $\mathbb{Q}$ and $[K:\mathbb{Q}]=n$. Let us call $\mathbb{O}_K$ ...
-2
votes
1
answer
186
views
An equality between $\pi$ and $\Gamma$ function [closed]
Consider the following equality:
$$\sum_{n=1}^{+\infty} (-1)^{n+1} \frac{(\frac{(2n-3)!!}{(2n-2)!!})^2*\frac{\pi}{2}}{n}=\frac{\Gamma(\frac{1}{4})^2}{2\sqrt{2\pi}}-\frac{2\sqrt{2}*\pi^{\frac{3}{2}}}{\...
- 171
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votes
0
answers
107
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Motivic $L$-functions came from automorphic representations
Langlands in his 1978 ICM talk made a conjecture that all motivic $L$-functions should arise as automorphic $L$-functions. A part of this conjecture, namely for some Hasse-Weil $\zeta$ functions is a ...
- 131
7
votes
1
answer
491
views
Suitable closed form for the A079501
Let $a(n)$ be A079501 (i.e., number of compositions of the integer $n$ with strictly smallest part in the first position).
The sequence begins with
$$
1, 1, 2, 2, 4, 5, 8, 12, 19, 28, 45, 70, 110, ...
- 3,464
17
votes
1
answer
1k
views
Can the Pythagorean Graph be finitely colored?
Define the Pythagorean Graph as having nodes $a,b\in \mathbb{N}_{\ge 3}$ and an edge $a\rightarrow b$ if and only if $a^2+b^2$ is a square. After much searching I found the example in the picture, ...
- 5,055
0
votes
0
answers
94
views
Relation between elements with fixed exponent over different $\mathbb{Z}^\times_p$
A primitive root $h$ of $n$ is a generator of the cyclic modulo multiplicative group $\mathbb{Z}^\times_n$.
Suppose, $\mathbb{P}_{\langle 2\rangle,N}=\{p_i <N\mid \langle 2\rangle=\mathbb{Z}^\...
1
vote
0
answers
114
views
Solution formula in an explicit equation over $\mathbb{F}_p^3$
I'm looking into a formula involving prime numbers $p \geq 7$ and an equation's solutions. The equation in question is:
$$z^2 = (x^2 - 4x)(y^2 - 4y)((x + 1 - y)^2 - 4x),$$
where $(x,y,z)\in \mathbb{F}...
- 71
1
vote
0
answers
79
views
How to prove this peculiar relationship between minimal polynomials of Ramanujan class invariants?
The Ramanujan class invariants (a.k.a. "Ramanujan-Weber class invariants")
are defined for $n>0$ by
$$G_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1+e^{-(2k+1)\pi\sqrt{n}}\right)...
- 13.2k
4
votes
1
answer
200
views
Fibonacci and matrix modular exponentiation
I'm interested in a few problems that are related enough that I decided to put them all in one question.
What are the fastest known algorithms for finding large Fibonacci numbers modulo $p^k$, and ...
- 437
2
votes
0
answers
203
views
Limits related to the floor function
Here I am still interested in the function $f(n,k)=\frac{2^{k}+1}{2^{n}+1}\left\lfloor \frac{2^{n}+1 }{2^{k}+1}\right\rfloor$ and a Tauberian property that I would like to check.
Let $\lambda>1$ be ...
- 275
2
votes
1
answer
254
views
On properties of sums involving the floor function
During my research on properties of fractional part and integer part functions, I was led to consider the function of two variables $f(n,k)=\frac{2^{k}+1}{2^{ n}+1}\left\lfloor \frac{2^{n}+1}{2^{k}+1}\...
- 275
1
vote
0
answers
68
views
A possible generalization of Brauer's theorem about the prime factors of the period and index of a central simple algebra
Let $K$ be an arbitrary field, and let $K^s$ be a fixed separable closure of $K$.
Let $F/K$ a be a finite Galois extension in $K^s$.
Let $n>0$ be a natural number.
Let $A$ be a central simple ...
- 13.6k
3
votes
1
answer
485
views
$\zeta(2n+1)$ - Is this formulation helpful?
Cross-posting alert: I posted This on MSE. I read through the guidelines for cross-posting on both the sites and my conclusion is that I am not violating any guidelines.
Briefly, I derived some ...
- 141
2
votes
1
answer
100
views
Recursion for the Chebyshev transform of $m^n$
Let
$$
R(n, q, m) = R(n-1, q+1, m) + \sum\limits_{j=0}^{q} (-1)^{q-j}R(n-1, j, m), \\
R(0, q, m) = (m-1)^q
$$
I conjecture that $R(n, 0, m)$ is a Chebyshev transform of $m^n$.
Examples of Chebyshev ...
- 3,464
3
votes
0
answers
152
views
A sharper estimate for a generalization of the sum-of-divisors function
I am interested in the function $f_n(m)$ which can be defined by the Dirichlet generating function $$\zeta(s)\zeta(s - 1) \cdots\zeta(s - n + 1) = \sum\limits_{m = 1}^\infty \frac{f_n(m)}{m^s} $$
This ...
- 31
3
votes
0
answers
99
views
Bounding $h_3(D)$ by number of points on an elliptic curve
According to Helfgott-Venkatesh, Let $E(D)$ denote the elliptic curve $y^2 = x^3 + D$, then $h_3(Q(\sqrt D))$, which is the 3-part of the class number of the Quadratic Field with discriminant $D$, or ...
- 51
2
votes
0
answers
56
views
Aligning frequencies
Let $\omega_1, \omega_2, \dots, \omega_n$ be frequencies between $1$ and $\log n$. I would like to find an upper bound for a point $t$ that align these frequencies up to a small error $\delta$, that ...
- 21
9
votes
2
answers
608
views
Another limit involving the fractional part
It is known that
$$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\left\{ \frac{n}{k}\right\} =1-\gamma$$
where $\left\{ x\right\}$ is the fractional part of $x$ and $\gamma$ is the Euler constant. ...
- 275
7
votes
1
answer
420
views
An asymptotic formula in Apéry's proof of the irrationality of $\zeta(3)$
Let $a_n$ be the Apéry sequence
$$
a_n = \sum_{0\leq k\leq n}\binom{n}{k}^2\binom{n+k}{k}^2.
$$
Reading the 1978 paper Démonstration de l’irrationalité de $\zeta(3)$ (d’après R. Apery) of Cohen, at ...
- 191
4
votes
0
answers
130
views
Is there a statement in Presburger arithmetic about primes this simple heuristic fails for?
I came up with the following conjecture while thinking about ways to formulate some heuristics about primes:
Conjecture: Given a statement $s$ in Presburger arithmetic, using an additional unary ...
- 2,979
3
votes
0
answers
113
views
Root separation for polynomials of bounded height
Consider integer polynomials $p$ of degree $\leq d$ and height $\leq H$, irreducible over $\mathbb{Q}$. The separation $\text{sep}(p)$ of $p$ is defined as the minimum absolute difference between any ...
- 31
5
votes
0
answers
167
views
Modularity lifting theorem à la Kisin
In its paper "Moduli of finite flat group scheme and modularity", Kisin showed the following theorem:
One of the main tool used is the scheme $\mathscr{GR}_{V_\bf F, \xi}$ defined in ...
- 61
2
votes
0
answers
154
views
Interesting conjecture by Sequence Machine
Let $a(n)$ be A344960 (i.e., position of binary complement of $n$-th word in A341258). By definition, in order to calculate $a(n)$, we need to know A341258. Below we will correspond this sequence with ...
- 3,464
4
votes
0
answers
741
views
One of the numbers $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational
I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive ...
- 1