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

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28 views

Number of zeros in Pascal-like triangles up to $2m+1$

Let $m \geqslant 2$ be a fixed integer. Then we have an integer sequence given by $$a(m)=\sum\limits_{n=0}^{2m+1}\sum\limits_{k=0}^{n}\operatorname{binmod}(n,k,m)$$ where $$\operatorname{binmod}(n,k,m)...
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23 views

"Residue-class generic" numbers

Let us call a set $D\subseteq\mathbb Z$ residue-class dense if for each residue class $[a]_n=\{kn+a\mid k\in\mathbb Z\}$, there is a residue class $[b]_m$ with $[b]_m\subseteq [a]_n\cap D$. Using the ...
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69 views

Mathematical theory of financial markets and implementations [closed]

Why are trading and financial markets not active areas of research in mathematics? Aren't there any applications of fields like probability theory, number theory, chas theory, etc to financial ...
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43 views

What is "the quotient of the universal ordinary Hecke algebra corresponding to an ordinary $\Lambda$-adic form"?

Let $\Lambda := \Bbb Z_p[[T]]$ be the usual Iwasawa algebra. In Jha and Sujatha - On the Hida deformations of fine Selmer groups on page 181, the authors refer to the quotient $\Bbb H^{\text{ord}}_{\...
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64 views

Literature on analogous arithmetic function of logarithm function

In number theoretical estimations, often we take the logarithms of a natural number to express it properly. A perfect example of this is the von-Mangoldt function. I am looking for an analogous ...
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2answers
210 views

Moebius function of finite abelian groups

I am wondering if there is any literature on general formula of the Moebius function of subgroup lattices of any finite abelian group $G$? What I know is When $G$ is cyclic, the Moebius function is ...
6
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1answer
139 views

Explicit estimates for $N(T,\chi)$ (not $N(T,\chi)+N(T,\overline{\chi})$)

Let $N(T,\chi)$ denote the number of zeros of $L(s,\chi)$ with imaginary part between $0$ and $T$, with any zero with imaginary part equal to $T$ or to $0$ (not that the latter kind really exists) ...
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1answer
93 views

Does Rankin-Selberg convolution preserve primitivity?

Call $L$-function any element of an L-rig (see Are there infinitely many L-rigs? for a definition). Suppose $F$ and $G$ are two primitive L-functions. Is $F\otimes G$ itself primitive? If yes, does ...
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129 views

Conditions under which an $\eta$-quotient becomes a **weak** modular form (reference request for theorems similar to Ligozat's theorem)

For any $z \in \mathcal{H}$, let $q = e^{2\pi iz}$; and the eta function is defined as ${\displaystyle \eta (q) =q^{\frac {1}{24}}\prod _{n=1}^{\infty }\left(1-q^{n}\right).}$ By an $\eta$-quotient ...
12
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1answer
394 views

Representing $x^3-2$ as a sum of two squares

Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers. Ideally, I am looking for a proof method that also applies for other $P(x)$, ...
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53 views

Conjugacy classes in normalized unit group of a group ring

Let $V(FA_4)$ be the normalized unit group of the group ring $FA_4$, where $F$ is the field containing 4 elements and $A_4$ is the alternating group on 4 symbols. How can I find conjugacy classes of ...
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111 views

Nearby cycle is tamely ramified?

Let $S$ be a henselization of a closed point $s$ in a smooth algebraic curve $C$ over some finite field $\mathbb{F}_q$. Then we can consider nearby cycles over $S$. Let $s$ be the closed point of $S$ ...
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1answer
226 views

Why do we need to represent integers as the sum of three cubes? [closed]

It is conjectured that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to $$ a^3+b^3+c^3=k. $$ Some cases for integer $k$ becomes too hard like $42$ which it ...
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184 views

Messing around with $e+\pi$

This question originates from the conjecture that $e+\pi$ is transcendental, and that $e$ is conjectured not to be a period. Jianming Wan in his paper Degrees of periods states that the transcendence ...
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84 views

Bounds on exponential and character sums of ratio of linear recurrences

Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\chi$ be a non-trivial additive character of $\mathbb{F}_q$, and let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$. Also,...
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1answer
164 views

Least number coprime to a given integer

For a positive integer $n$ let $$f(n):=\min\{m\in \mathbb N: m>1, \gcd(m,n)=1\} .$$ Equivalently, $f(n) $ is the smallest prime not dividing $n$. Is there any upper bound literature for this? It is ...
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71 views

On $g^{2^x}=B$ modulo $p$

Let $p$ be prime and $g$ positive integer. Define $f(g,x)=g^{2^x} \bmod p$. Q1 Given $g,p,B=f(g,X)$, what is the complexity of finding $X$? If necessary assume $\varphi(p-1)$ is smooth. Some ...
6
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1answer
199 views

Another generalization of parity of Catalan numbers

Recently, a question by T. Amdeberhan gathered up many enjoyable proofs that a Catalan number $C_n$ is odd if and only if $n=2^r-1$. Noam D. Elkies' answer considered $F=\sum_{n=0}^\infty C_n x^{n+1}$....
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125 views

Conditional results on average size of Mertens' function

Let $M(x) = \sum_{n \le x} \mu(n)$ where $\mu$ is the Möbius function. Titchmarsh, in his book on the Riemann zeta function, considers consequences of the hypothesis that $$\int_{1}^{X} \left( \frac{M(...
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0answers
121 views

Exactness of a term after taking Pontryagin dual: a step in the proof of Poitou-Tate duality

I'm reading the proof of Poitou-Tate duality in the book Galois Cohomology and Class Field Theory by David Harari. After some arguments, we get a exact sequence $$ \mathbf{P}^1_S(k,M^{'})^* \...
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170 views

Galois group of a period

Following Jianming Wan's paper entitled "Degrees of periods", can one define the Galois group of a period as the maximal group of permutations of the variables appearing in the expression of ...
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1answer
129 views

On some claims on cyclic modules over Hecke algebra used in Serre's "Quelques applications du théorème de densité de Chebotarev"

I have been reading section 7 of Serre's "Quelques applications du théorème de densité de Chebotarev" (http://www.numdam.org/item/PMIHES_1981__54__123_0/), and in particular have been trying ...
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0answers
357 views

Karatsuba photo [closed]

Can anyone confirm if the following link displays a photo of A. A. Karatsuba? https://commons.m.wikimedia.org/wiki/File:A.A.Karatsuba_in_Crimea.jpg
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110 views

Elliptic curves and localizations at various primes

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime at which $E$ has good reduction. Let $D=D_{E,p}$ be the $p$-torsion in the cokernel of the map $E(\mathbb{Q})\otimes\mathbb{Z}_p\...
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116 views

Can a lower bound for this weakening of Goldbach's conjecture be reached?

Say a non negative integer $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are prime, and that a non negative integer $w$ is a weak primality radius of $m$ if $\omega(m-w)=\omega(m+w)=1$, ...
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1answer
148 views

Are Li's numbers $\lambda_n$ absolutely convergent for $n>1$?

Li's numbers $\{\lambda_n\}$ are defined as $$\lambda_n=\frac{1}{(n-1)!}\frac{d^n}{ds^n} [s^{n-1}\log\xi(s)]_{s=1} $$ for all positive integers $n$. Also $\lambda_n$ is given as a sum over the non ...
2
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58 views

Correspondence between class group of binary quadratic forms and the narrow class group via Dirichlet composition: an elementary approach?

I have been trying to explore and learn about connections between the form class group and the ideal class group. To be on the same page, we define the form class group of a negative discriminant $D \...
2
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0answers
191 views

Two conjectures about generalised A329369

Let $m \geqslant 2$ be a fixed integer. Let $$\operatorname{wt}(n,m)=\operatorname{wt}\left(\left\lfloor\frac{n}{m}\right\rfloor,m\right)+n\bmod m, \operatorname{wt}(0,m)=0$$ Then we have an integer ...
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0answers
91 views

Can PARI compute class numbers without factoring the discriminant?

When calculating properties of algebraic number fields, one of the hardest steps is factorizing the discriminant of a defining polynomial. This is necessary in the Pohst-Zassenhaus algorithm for ...
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0answers
175 views

Ambiguity about the exact definition of coefficients of modular forms

You can see the parts after my questions in the boxes. I received the answer to my first question in the comments. I am confused about the definition of $a_n$ and $b_n$ in Part II below. I know the ...
3
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0answers
104 views

Global Vogan A-packet is infinite set?

For an cuspidal automorphic representation of general linear group, we can attach its global Vogan A-packet. Though I thought that it is finite set, in some paper, it is written that there are ...
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0answers
160 views

A paper 'the rank of elliptic curves' by Brumer [closed]

I want to download a paper: Brumer, Armand; Kramer, Kenneth The rank of elliptic curves. Duke Math. J., but I have no access to Duke journal. I want to read the paper for two years. Please help me!
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192 views

The Tate-Nakayama theorem and inflation

Let $K$ be a nonarchimedean local field, and $L/K$ be a finite Galois extension with Galois group $G={\rm Gal}(L/K)$ of order $n=[L:K]$. By local class field theory, there is a canonical isomorphism $$...
2
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1answer
274 views

Tannakian fundamental group of automorphic representations

Let $\mathcal{C}_{\mathrm{aut}}(G, F)$ be the category of automorphic representations of a connected reductive group $G$ over a number field $F$. If this is a Tannakian category, it has an associated ...
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99 views

Finding elements in a real extension of $\mathbb{Q}$ that are close to some number in $\mathbb{R}$

Let's consider a set of numbers that one knows to high precision, and one knows or has a strong suspicion that `exact versions of these numbers' (see below) belong to a real extension of $\mathbb{Q}$. ...
4
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130 views

Magic hourglass of squares hyperelliptic equation

I have been looking into the problem of the magic square of squares, or more specifically, the magic hourglass of squares, like so: $a^2$ $b^2$ $c^2$ $ $ $ $ $ $ $ $ $ $ $d^2$ $e^2$ $f^2$ $g^2$ ...
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1answer
82 views

Modulo $2$ binomial transform of A243499 applied $k$ times

Let $m \geqslant 1$ be a fixed integer. Let $f(n)$ be A007814, exponent of highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. ...
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50 views

Modulo $2$ binomial transform of A124758

Let $f(n)$ be A153733, remove all trailing ones in binary representation of $n$. Here \begin{align} f(2n)& = 2n\\ f(2n+1)& = f(n)\\ \end{align} Then we have an integer sequence given by \begin{...
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1answer
251 views

Factorial primes: expected finite or infinite?

A factorial prime is of the form $n! \pm 1$. The first $14$ factorial primes are listed in the Online Integer Sequences (OEIS) A088054: $$ 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, ...
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0answers
180 views

Can $\exp(W(\sqrt{\ln(\sqrt{n})})$ be an integer?

Let $W(z)$ be the Lambert $W$ function and $n$ a positive integer. Is it possible that $\exp(W(\sqrt{\ln(\sqrt{n})})$ is an integer? If $\exp(W(z))$ is an integer, say $k$, then we get $\frac{z}{k}=W(...
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2answers
573 views

Embedding torsors of elliptic curves into projective space

Suppose I have a genus 1 curve $C$ over a field $k$. If $C$ has a point, then we can embed it into the projective plane by a Weierstrass equation. Now let us suppose that $C$ does not have a point (so ...
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0answers
49 views

Inverse modulo $2$ binomial transform of generalised A284005

Let $m \geqslant 1$ be a fixed integer. Let $\operatorname{wt}(n)$ be A000120, $1$'s-counting sequence: number of $1$'s in binary expansion of $n$ (or the binary weight of $n$). Let $f(n)$ be A007814, ...
11
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0answers
288 views

List of problems that Erdős offered money for?

Is there a list somewhere of all the problems that Erdős offered cash awards for, including both solved and unsolved problems? One would think that the answer is yes, but so far I have had no luck ...
11
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1answer
741 views

Is the divisibility graph of the proper divisors of n more often planar than not?

Define the divisibility graph of a set of positive integers as the graph whose vertices are the integers, two of which are joined by an edge if one divides the other. For all N, is it true that ...
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0answers
85 views

Subsequence which is identical to A122778

Let $\operatorname{wt}(n)$ is A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$). Let $a(n)$ be A284005, \begin{align} a(0)& = 1\\ a(n)& = (1+\operatorname{wt}(n)...
3
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0answers
210 views

Rational points on surfaces

Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form $$ S = \{a_0(t)x^2+a_1(t)xy+a_2(t)x+a_3(t)y^2+a_4(t)y+a_5(t) = 0\} $$ where $...
14
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4answers
2k views

Collecting alternative proofs for the oddity of Catalan

Consider the ubiquitous Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. In this post, I am looking for your help in my attempt to collect alternative proofs of the following fact: $C_n$ is odd if and ...
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0answers
131 views

Open tours by a biased rook (proof verification)

Related questions: Number of open tours by a biased rook on a specific $f(n)\times 1$ board which end on a $k$-th cell from the right Sum with products turned into subsequences Combinatorial ...
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0answers
100 views

Inflation in degrees $0$, $-1$, and $-2$ for Tate cohomology of finite groups

Let $\pi\colon G'\to G$ be a surjective homomorphism of finite groups, and let $A$ be a $G$-module. I need explicit formulas for the inflation maps $${\rm Inf}^{r}\colon H^{r}(G,A)\to H^{r}(G',A)$$ ...
-1
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1answer
465 views

Can you explain this weird pattern in Collatz conjecture? [closed]

Extreme disproportion in the dispersion of "Digital Roots" of the highest numbers reach in the Collatz Conjecture. I calculated the Digital Root remainder mod 9 for the highest numbers ...

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