# 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

15,946
questions

3
votes

0
answers

69
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.,...

0
votes

0
answers

40
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

67
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

210
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 ...

2
votes

1
answer

91
views

### 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.
$abc$ ...

2
votes

0
answers

115
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 $N$ ...

4
votes

1
answer

160
views

### 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 ...

0
votes

0
answers

165
views

### 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
$$
\...

0
votes

0
answers

78
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.

15
votes

0
answers

288
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 ...

2
votes

0
answers

112
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 ...

6
votes

1
answer

296
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.

2
votes

0
answers

200
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 ...

2
votes

1
answer

136
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 ...

18
votes

0
answers

521
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
vote

0
answers

68
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 ...

1
vote

0
answers

61
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
...

3
votes

2
answers

203
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{\...

0
votes

1
answer

97
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 $...

-3
votes

0
answers

63
views

### 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 $\mathbb{O}_K$ ...

-1
votes

1
answer

179
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}}}{\...

3
votes

0
answers

105
views

### 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 ...

7
votes

1
answer

489
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, ...

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, ...

0
votes

0
answers

93
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

113
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}...

1
vote

0
answers

69
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)...

4
votes

1
answer

197
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 ...

2
votes

0
answers

181
views

+150

### 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 ...

-2
votes

0
answers

166
views

### Approximation of $\pi$ [closed]

I found a relation that gives $\pi$:
$$\sum_{n=0}^{\infty} \frac{(2n)!}{2^{4n+1}(n!)^2(2n+1)}=\frac{\pi}{6}$$
To prove this formula, can we use the Maclaurin series?
Thank you.
EDIT: here is a second ...

2
votes

1
answer

252
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}\...

1
vote

0
answers

67
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 ...

3
votes

1
answer

476
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 ...

2
votes

1
answer

99
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
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 ...

3
votes

0
answers

96
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 ...

2
votes

0
answers

55
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 ...

9
votes

2
answers

605
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. ...

7
votes

1
answer

416
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 ...

4
votes

0
answers

129
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 ...

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 ...

5
votes

0
answers

163
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 ...

2
votes

0
answers

153
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 ...

4
votes

0
answers

740
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 ...

7
votes

0
answers

3k
views

### Is there a mistake in Mochizuki's proof of Theorem 1.10 in IUTT IV? [closed]

In Global character of ABC/Szpiro inequalities, Peter Scholze says that he thinks Joshi's proof of the abc conjecture in his paper has a mistake in Proposition 6.10.7. However, for the proof of ...

4
votes

1
answer

168
views

### Density of automatic sets recognized by certain automata with sink states

I want to know about the density of automatic sets where the DFA recognizing it has a particular nice form. I'm going to start with a simple version of the question and then add complications until we ...

3
votes

1
answer

300
views

### On the equation $7x^3 + 2y^3 = 3z^2 + 1$

The question is whether there exist integers $x,y,z$ such that
$$
7x^3+2y^3=3z^2+1.
$$
After a similar equation On the equation $9x^3+y^3=z^2+3$ has been solved, this is one of the nicest cubic ...

16
votes

1
answer

531
views

### Limit involving the fractional part and the Fibonacci numbers

Helo,
Let $F(n)$ be the $n$th Fibonacci number, if $\left\{ x\right\}$ denotes the fractional part of $x$, how proving
$$\lim_{n\rightarrow\infty}\frac{1}{2n}\sum_{k=1}^{2n}\left\{ \frac{F(2n)}{F(k)}\...

5
votes

1
answer

684
views

### Geometric mean of prime factors of all numbers up to n

Through numerical calculations I have discovered that for any natural number $n \geq 2$, the geometric mean of the prime factors of all natural numbers $\leq n$ can be approximated well by $1.6653 \...

4
votes

1
answer

205
views

### Conditional convergence of Artin $L$-functions

Let $k$ be a number field and $V$ a non-trivial irreducible Artin representation over $k$. Consider the associated Artin $L$-function with corresponding Euler product decomposition $L(V,s)= \prod_v ...