# 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

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

14,558
questions

2
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Let $K$ be a non-archimedean local field of characteristic 0. Fix a uniformiser $\pi$ and an algebraic closure $\bar{K}$. The theory of Lubin--Tate formal groups gives an explicit construction of the ...

6
votes

1
answer

727
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This is a refinement of my question asked earlier, which is answered beautifully in the negative by Thomas Browning. The example he gave was geometrically reducible. Now I want to ask the same ...

2
votes

0
answers

77
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Let‘s take $G=GL_n$ over a number field $F$ for example.
It's already known that every irreducible automorphic representation $\pi$ is a irreducible component of a induced representation $I(G,P;\...

-9
votes

0
answers

76
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I am just an amateur. Usually proving something comes down to overcoming a finite number of obstructions. Typically a handful.
I wonder if there could be specific structure or absence of specific ...

23
votes

1
answer

1k
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Does there exist a polynomial $f \in \mathbb{Z}[x,y]$ such that
$$\displaystyle f(a,b) > 0 \text{ for all } a,b \in \mathbb{Z}$$
and
$$\displaystyle \liminf_{(x,y) \in \mathbb{R}^2} f(x,y) = -\...

9
votes

3
answers

359
views

I'm looking into expanding my knowledge in ways to show that some constant is irrational. I'm gonna give some examples of irrationality proofs, and I'm interested in learning what strategies you guys ...

4
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0
answers

104
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Recently I read Prof. Cogdell's notes: Lectures on L-functions, Converse Theorems, and Functoriality for $GL_n$. (Co)
In chap.2.3, the conception of smooth automorphic forms is introdued. Explicitly, ...

0
votes

0
answers

69
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Integer $n$ is Wieferich number if $2^{\phi(n)}-1 \equiv 0 \pmod {n^2}$.
Wieferich prime is Wieferich number with $n$ prime.
It is an open problem if there are infinitely many Wieferich primes
and ...

4
votes

1
answer

292
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I suspect that certain problems regarding the base 10 representation of natural numbers may be undecidable simply because there's no way to even start.
Take any exponentially growing sequence like $16^...

2
votes

1
answer

264
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The truncated explicit formula has the shape
\begin{equation}\label{1}
\psi(x) =x-\frac{\zeta^{\prime}(0)}{\zeta(0)}-\sum_{|\rho|\leq T}\frac{x^{\rho}}{\rho}+\sum_{n=1}^{\infty}\frac{x^{-2n}}{2n}+...

2
votes

0
answers

108
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In 1998 John Friedlander and Henryk Iwaniec famously proved the asymptotic formula
$$\displaystyle \mathop{\sum \sum}_{a^2 + b^4 \leq x}\Lambda(a^2 + b^4) = \frac{4x^{\frac{3}{4}}}{\pi} \int_0^1 (1 - ...

39
votes

2
answers

6k
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Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635
In that preprint, Kirti Joshi claims that
he agrees with Scholze and ...

2
votes

0
answers

66
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Working on suitable closed form for A329369, I discovered very useful coefficients, which have the following recurrence relation:
$$T(0,1)=T(0,2)=1$$
$$T(n,1)=1, n>0$$
$$T(0,k)=0, k>2$$
$$T(2n+1,...

7
votes

1
answer

538
views

Let $^na$ denote the $n$th tetration of $a$, so that $^0a=1$ and
$$^{n+1}a=a^{^na}$$
for $n=0,1,\dots$. (For complex $x$ and $y$, here we use the definition $x^y:=e^{y\ln x}$, where $\ln$ is the ...

2
votes

0
answers

93
views

Let $d$ be a positive integer. Let $f(d,a)$ be the number of values of $x$ in $[1,d]$ such that $$x^{a}\equiv 1\pmod{d}.$$ I wanted to know if for some $0<\epsilon<1$, we can prove the following ...

2
votes

1
answer

186
views

How many prime numbers of $b$ bits are there?
Beyond the prime number theorem, one can give explicit bounds on the number of primes below some integer $n$, or in a given interval. For instance, Rosser ...

4
votes

1
answer

186
views

Problem: classify all pairs $(k,P)$ such that $P(x)$ divides
$$x^kP(x+1)+(x-1)^kP(x-1),$$ where $k\ge4$ is an integer, and $P$ a nonconstant monic
polynomial with rational coefficients.
I have found ...

3
votes

1
answer

241
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This question is based on this Math StackExchange question by user buja and my related Math StackExchange question which was asked about a month ago and is still unanswered.
I believe
$$\phi_2(n)=n \...

1
vote

1
answer

142
views

Let $F$ be a nonarchimedean local field, say, charactersitic $0$. Is there any general theorem that tells when $\sqrt{-1}$ exists in $F$? How often does it happen?

0
votes

0
answers

198
views

Let $a_n$ be an integer sequence where
$$
a_n=4n^6 + 4n^5 - 11n^4 + 7n^2 - 4n + 1
$$
Prove or disprove that no member of the sequence is a perfect square (except $a_0$ and $a_1$).
My computer can't ...

1
vote

1
answer

144
views

Let $(a_0,a_1,\dotsc)$ be an infinite sequence of natural numbers containing all the natural numbers. Assume that $a_x\equiv a_y\pmod{m}$ if and only if $x\equiv y\pmod{m}$, for all $x,y\in\mathbb{N}$ ...

1
vote

0
answers

87
views

Let $q(n)$ be A007814, i.e., the number of trailing zeros in the binary representation of $n$. Here
$$q(2n+1)=0, q(2n)=q(n)+1$$
Let $a(n)$ be A329369. Here
$$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(...

0
votes

0
answers

39
views

Let us assume we have a convolution of g-copies of a summation and j-copies of another summation, and they are multiplied together, so we have a product of $g+j$ summations. And $g+j=u$.
Series1:
$
\...

4
votes

0
answers

121
views

This question concerns the asymptotic maximum of $L(1,\chi_d)/ \log \log d$ when
$d$ is even. I computed it for even $d$ less than 7.5 million. The largest
value is $1.230126$, occurring for $d= ...

3
votes

1
answer

276
views

Suggested by this problem:
Do the sets of all odious / evel numbers meet every infinite arithmetic progression?
A number is odious if it contains an odd number of digits $1$ in its binary ...

0
votes

1
answer

93
views

Let $\varphi$ denote the Euler's totient function. Is there any reference in literature for the value of sum $$\sum_{\substack{r\le x\\ d\mid r}}\gcd(\phi(d),r)$$ where $d$ is some fixed positive ...

4
votes

0
answers

84
views

Let $n$ be a positive integer and $\mathrm{ord}_{n}(a)$ be the least positive integer $d$ such that $n\mid a^{d}-1$. I wanted to know if for some choice of $y=y(x)$, one can obtain asymptotics for ...

-1
votes

0
answers

86
views

Let $p$ be a prime and let $x$ be a positive integer. How do I prove that the equation
$$y^2=px^3+p$$
has infinitely many integer solutions? I tried to tackle this problem using elliptic curves and ...

2
votes

1
answer

131
views

In an attempt to solve an unrelated problem, I was led to the task of estimating/bounding from above sums of the form
$$\sum_{m=1}^\infty\lambda(m)e\left(-\frac{am}{q}\right)h(m)$$
where $\sum_{m=1}^\...

2
votes

0
answers

98
views

Let $\chi$ be a non-trivial multiplicative character of the $3$-adic field $K:=\mathbb{Q}_3$. I would like to find out the relation between the conductor of $\chi$ and the conductor of $\chi^3$.
Let ...

21
votes

0
answers

6k
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Now that a week has passed since Zhang posted his preprint Discrete mean estimates and the Landau–Siegel zero on the arXiv, I'm wondering if someone can give a high-level overview of his strategy. In ...

0
votes

0
answers

126
views

I am a self studying student and I am interesting with diophantine equations. I have the following questions:
How can I know that a diophantine equation is solved or not?
the second question is what ...

1
vote

1
answer

227
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We are given $1\leq S \leq 10^9$ and $1 \leq P \leq 10^9$.
We need to pick $k$ integers $x_1, x_2, \dotsc, x_k$ (all of which have to be $>1$) such that $\sum_k{x_k}=S$ and $\prod_k{x_k}=P$.
What ...

2
votes

0
answers

238
views

NOTE: I've edited the question one last time, to be much simpler, in the hopes of getting more responses.
SETUP: Let $p_n$ denote the $n$th prime, let $p_x = p_{\lceil x \rceil}$ for all $x > 0$, ...

2
votes

1
answer

99
views

For a given integer $n$, I am interested in the number of different numerical semigroups one can make with a generating set consisting only of integers in $[n]$.
I have done some small examples. For $...

2
votes

0
answers

205
views

Consider a quadratic field ${\mathbb Q}(\sqrt d)$ and its
$L$-function, evaluated at 1. Littlewood proved a famous bound (under GRH) and it's known that for infinitely many $d$, half this bound is ...

1
vote

0
answers

236
views

Let $f_1, \dots, f_n$ be a finite set of polynomials in the polynomial ring $Z[x_1, \dots, x_m]$. At a prime $p$, let $N_p$ be the number of solutions $x=(x_1, \dots, x_m)\in (\mathbb{Z}/p\mathbb{Z})^...

3
votes

0
answers

166
views

Does there exist an algorithm or something of the sort to reverse-engineer a curve from its modular form (weight two eigenform with complex coefficients)? I am aware that sometimes there isn’t a ...

0
votes

0
answers

69
views

Let $\Lambda$ be a lattice in $R^n$ and $R>0$ a real number.
Consider the number $N$ of points in $\Lambda$ of norm less than $R$. Let $R$ goes to infinity. What can be said about the asymptotic ...

3
votes

0
answers

144
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I recently learned Sharifi's conjecture in this article by F.Takako and K.Kato.
According to my understanding, this conjecture states that (the conjugation invariant) of the projective limit $\...

2
votes

0
answers

2k
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There are some integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. One of them is the integral introduced by Selberg related to estimating the variance of primes in ...

2
votes

0
answers

102
views

Let $K$ be a finite extension of $\mathbb{Q}$ of degree $d$ and let $E(K)$ be an elliptic curve over the field $K$ with coefficients in $K$. Let us fix $d$ and vary over all the possible $K$, in turn ...

1
vote

0
answers

53
views

In the case of complex elliptic surfaces we can consider the elliptic modular surfaces, as Shioda did in his paper "On elliptic modular surfaces". The latter have Mordell-Weil rank 0.
Is ...

8
votes

1
answer

1k
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For a natural number not a perfect square, is there always at least a prime number for which it is a primitive root?
Artin's conjecture on primitive roots is that there are infinitely many such primes....

5
votes

1
answer

217
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It is known that the number of ways to express $n \in \mathbb{Z}_{\geq 0}$ as a sum of three squares (let's denote it as $r_3(n)$) can be expressed as Hurwitz-Kronecker class number (certain weighted ...

2
votes

0
answers

71
views

Let $n \in \mathbb{N}$ be a positive integer.
It is possible to prove that the equation $F_1(n)=m$ where $m \in \mathbb{Z}$ and
$$
F_1(n)=(1+2+\ldots+n)\cdot\left(\frac{1}{1}+\frac{1}{2}+\dots+\frac{1}...

1
vote

0
answers

71
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Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers
and $S$ a finite subset of the real places. Let $\mathfrak{m}
\subset \mathcal{O}_K$ an ideal. The ideal
theoretic
ray class group of $\...

8
votes

1
answer

3k
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Zhang 2022 proves a somewhat suspicious formula:
$$L(1,\chi) \gg (\log D)^{-2022}$$
This raises the obvious-but-frivolous question: did he intentionally weaken the constant to get the current year?

0
votes

0
answers

90
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Let $q(n)$ be A007814, i.e., number of trailing zeros in the binary representation of $n$. Here
$$q(2n+1)=0, q(2n)=q(n)+1$$
Let $a(n)$ be A329369. Here
$$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(2n-2^{...

9
votes

0
answers

85
views

$\DeclareMathOperator\SL{SL}$Let $p$ be a prime, and let $\Gamma_r$ denote the kernel of the map $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p^r\mathbb{Z}_p)$.
Explicit formulas with formal ...