# 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

4
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1
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290
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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^...

7
votes

1
answer

520
views

Let $\pi$ and $\pi'$ be two general automorphic representation on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$.
I heard that the rankin-selberg convolution L-function $L(s,\pi\...

9
votes

3
answers

357
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 ...

92
votes

15
answers

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I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...

11
votes

1
answer

679
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Let $k$ be a positive integer and let $p$ be a prime. In my 2011 PAMS paper joint with my former student W. Zhang [Proc. Amer. Math. Soc. 139(2011), 1569-1577], we studied when $$S(k)=\left\{\binom nk:...

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}+...

5
votes

2
answers

245
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I am a little confused on the $p$-adic regulator on elliptic curves and what happens when you switch to different Weierstrass models. Restrict to ell. curves over $\mathbb Q$ for simplicity.
From my ...

2
votes

0
answers

70
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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 ...

9
votes

2
answers

416
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(What follows is motivated by an answer to Fourier optimization problem related to the Prime Number Theorem)
Let $f:\mathbb{R}\to [0,\infty)$ be such that
(a) $\int_{\mathbb{R}} f(x) dx = 1$,
(b) $\...

39
votes

2
answers

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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 ...

35
votes

2
answers

11k
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Hi,
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] ...

6
votes

1
answer

726
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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;\...

2
votes

1
answer

131
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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}^\...

-9
votes

0
answers

76
views

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 ...

4
votes

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, ...

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) = -\...

24
votes

1
answer

1k
views

Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...

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 ...

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 \...

2
votes

0
answers

108
views

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 - ...

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

66
views

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,...

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 ...

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 ...

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 ...

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 ...

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

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?

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

90
views

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^{...

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:
$
\...

0
votes

1
answer

93
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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 ...

12
votes

1
answer

2k
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Let $P(x;a,b) := \{an+b, 0\leq n \leq x \} $ denote an arithmetic progression. Further let $A(x;a,b)$ denote the number of elements of $P(x;a,b)$ that are squares. It's an old conjecture of Rudin ...

4
votes

0
answers

121
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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 ...

4
votes

0
answers

84
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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

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 $x > 0$, ...

4
votes

0
answers

341
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I just watched Kannan Soundararajan's talk on the distributions of valus of zeta and $L$-functions at virtual ICM 2022. In his talk, he introduced a theorem on Ramanujan's ternary form $\phi_{1}: x^{2}...

8
votes

3
answers

2k
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Hello!
Let $K$ be a number field. All abelian unramified extensions are contained in the Hilbert class field which is a finite extension 'maximal' with respect to this property. For general ...

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

98
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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 ...

1
vote

1
answer

130
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In the answer to this previous question , Noam D. Elkies proved that for any integer $x$, $x^3-x^2-2x+1$ can only have a divisors equal to $-1$, $0$, or $1$ modulo $7$. I would like to know what is ...

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 ...

8
votes

1
answer

459
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For integers $s \geq 0$ and $n \geq 1$ let $\sigma_s(n) := 1^s + \dotsb + n^s$ and let
\begin{equation} \sigma_s^*(n) \ : = \ \sum_{\substack{\scriptstyle 1 \, \leq \, m \, \leq \, n \\ \gcd(m,n) = 1}}...

3
votes

2
answers

384
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Is there any result known for the number of different solutions of $1 = \sum\limits_{k=0}^n \frac{1}{x_k}$ in dependency of the length $n$ of this partition?
All I know, up to now, is that there are ...

26
votes

2
answers

1k
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If Artin's conjecture on primitive roots is true, then 2 generates $(\mathbb{Z}/p\mathbb{Z})^\times$ for infinitely many primes $p$. Can one at least show that $(\mathbb{Z}/p\mathbb{Z})^\times$ is ...

52
votes

2
answers

8k
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This question is related to this previous question where I asked about ordinary Fourier coefficients.
##Special case: is Möbius nearly orthogonal to Morse
!
Harold Calvin Marston Morse (24 March 1892 ...