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

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44 views

Relative Leopoldt defect

Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$. Is there a bound of the Leopoldt defect of $M$ ...
1
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0answers
58 views

The Linnik problem for dimension $2$

For $N$ an integer, let $$\Omega_N:=\left\{\frac{\alpha}{\|\alpha\|}|\alpha \in\textbf{Z}^n~\text{and}~\|\alpha\|^2=N\right\}.$$ For $n=3$, Linnik asked if the set $\Omega_N$ was uniformly distributed ...
2
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0answers
48 views

On the size of residue class

Let $n \in \mathbb{N}$ be a odd number. Let $S \subseteq \{1,3,5,7,...,n-2,n\}$ and $|S|$ is even number. Let $R_i^k=\{a \mid a \in S \text{ } \&\text{ } a\equiv i \text{ }(mod \text{ } k)\}$ ...
1
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0answers
131 views

Computational number theory

Suppose that $p$ is prime and $q$ is an even number divides $p-1$, such that $q<\frac{p-1}{q}$ and $u$ has order $q$ modulo $p$. Let $S$ be the subgroup of $Z^*_p$ consisting of the powers of $u$. ...
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71 views

Considering a matrix with integrer entries over $\mathbb{Z}/p \mathbb{Z}$, does it remain full rank? [on hold]

Suppose I have an $m \times n$ matrix $M$ with integer coefficients, and suppose it has full rank. Let $p$ be a prime and now consider the matrix $\bar{M}$ over $\mathbb{Z}/p \mathbb{Z}$. Is it true ...
12
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1answer
245 views

Does Littlewood's bound on $\zeta(1+it)$ extend to all the partial sums?

Littlewood established that $2e^{\gamma} \geq \limsup_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be ...
10
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1answer
236 views

Polylogarithm sheaves

In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: $\mathrm{...
4
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1answer
151 views

Exceptional isomorphisms between finite simple Chevalley groups

Steinberg's "Lectures on Chevalley Groups" https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf contain ``a complete list of isomorphisms" among the various finite simple Chevalley ...
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56 views

On semi-complete ring K[X_1,X_2,…,X_∞]] and Popescu theorem

Let $P_n \colon= K[X_1,...,X_n]$ be a $n$-variables polynomial ring. We define 'semi-complete' polynomial ring $P_{\infty}$ by the following$\colon$ $P_{\infty} = K[X_1,...,X_\infty]] \colon = \...
2
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1answer
132 views

Criteria for the surjectivity of the reduction map of the $SL_n$-group scheme

Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map $$ \pi:SL_n(R)\rightarrow SL_n(R/I) $$ (In the original question I had put $GL_n$ instead of $SL_n$ ...
4
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1answer
127 views

Hilbert modular forms twist-equivalent to their conjugates

Let $L / K$ be a solvable (or cyclic) Galois extension of totally real fields, and let $f$ be a Hilbert modular newform over $L$. Suppose that, for every $\sigma \in Gal(L / K)$, the conjugate ...
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0answers
98 views

Combinatorial splitting in number rings

The goal of this problem is to see if there is a structured way to factor numbers constructed from a set of distinct odd primes $p_1$ through $p_n$ in a number ring. Take an arbitrary non empty ...
3
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3answers
311 views

Jacobi's theorem on sums of two squares (reference request)

One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals $$4(d_1(n)-d_3(n)),$$ where the function $d_i$ counts the number ...
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0answers
77 views

A basic Query regarding Riemann Zeta function [on hold]

The Euler Definition of Zeta is given as (extreme right): Using only Euler's definition, how can any value of s (real/complex) lead to the function being 0. As, no matter what the denominator of ...
14
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2answers
340 views

Can you use Chevalley‒Warning to prove existence of a solution?

Recall the Chevalley‒Warning theorem: Theorem. Let $f_1, \ldots, f_r \in \mathbb F_q[x_1,\ldots,x_n]$ be polynomials of degrees $d_1, \ldots, d_r$. If $$d_1 + \ldots + d_r < n,$$ then the ...
2
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3answers
267 views

Geometry of numbers argument: counting integers with some linear condition

I am interested in the proof of the following result: Suppose that $A > 1$, $\lambda \in \mathbb{R}$, and for $0 < Z \leq 1$, let $U(Z)$ be the number of integer solutions $v$ of \begin{...
11
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4answers
640 views

Smallest solution to $x^2 \equiv x\pmod{n}$

Given $n$, is it possible to upper bound the smallest $x > 1$ that satisfies the congruence $x^2 \equiv x\pmod{n}$? Obviously when $n$ is a prime power $x = n$, and we are in the worst situation. ...
1
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0answers
72 views

Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$

Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let $$ \mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} . $$ We ...
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0answers
108 views

Troost-Bourget identity $ N \sum_{d|N} 1 = \sum_{d| N} \sum_{l=1}^d \mathrm{gcd}(d,l) $ [on hold]

In the process of evaluating a "supersymmetric index", Bourget and Troost establish a rather elementary identity: $$ \frac{N}{m} \sum_{d| N} \sum_{l=1}^{\mathrm{gcd}(d,m)} \mathrm{gcd}\left[ \mathrm{...
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0answers
70 views

Sum-free sets of powerful numbers

For $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ with distinct primes $p_i$, call $\alpha= (\alpha_1,\dots,\alpha_r)$ the type of $n$ and denote by $N_\alpha$ the set of all naturals of this type. We ...
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88 views

A question about arithmetic progressions and prime numbers

"I took number $3$ and observed: $3$ is an arithmetic progression of length one. $3,5$ is an arithmetic progression of length two. $3,5,7$ is an arithmetic progression of length three. Then I took ...
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1answer
114 views

How to count fixed-sized subsets of pairwise co-prime numbers less than a prime, satisfying an additional ‎constraint‎?

In part of my research, I need to count (or find a sharp bound for) the number of ‎possible ‎ways to select $n$ distinct integers less than the prime $p$, say $r_1, r_2, …, r_n$, which ‎are pairwise ...
12
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2answers
592 views

Congruence equation and quadratic residue

The following observation makes me quite confused when I am trying to count the number of solutions of the equation: $$\sum_{k=0}^{M}{M \choose k}^2x^k=0$$ on finite fields $\mathbb{F}_p$ with the ...
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1answer
101 views

Is a positive integer determined by its sequence of typical primality radii?

This question is a follow-up to About Goldbach's conjecture . Assuming the truth of Goldbach's conjecture, suppose $n$ and $m$ are two positive integers such that $N_{2}(n)=N_{2}(m)=:N$ and that ...
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0answers
66 views

On the existence of certain type of arithmetic progressions [closed]

I asked this question on math.stackexchange about 4 hours ago and did not receive neither a comment or an answer. Maybe I am a little impatient and should have waited for some time to see will I ...
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0answers
48 views

Solution to congruency modulo p^e [closed]

How many solution are there for z which satisfies z^2 = p^f (mod p^e). z is element of Zp^e, f is even integer less than e, e is positive integer, and p is prime.
5
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1answer
157 views

A $p$-adic sum of reciprocals of powers

Let $p$ be a prime number and $k\geq 2$ an even integer. Consider the following $p$-adic integer: $$ S_{p,k} := \lim_{r\to+\infty} \sum_{a=1}^{p^r} \big(\frac{p^r}{a}\big)^k $$ Convergence is easy to ...
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1answer
139 views

Complete subring of F_p[[X]]

Pointed out on famous disbelief, I know now that there is an embedding $\iota_n \colon {\Bbb F}_p[[T_1,...,T_n]] \hookrightarrow {\Bbb F}_p[[X,Y]]$ for any $n \leq \infty$. Then I would like to ask ...
5
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0answers
135 views

Lifting points via étale morphism of adic spaces

This question was suggested to me during the reading of Huber's book about Etale Cohomology of Adic Spaces. I formulate this question here in the context of adic spaces, but I think, since a morphism ...
5
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1answer
244 views

Infinitely many primes coming from Euclid's proof

When teaching Euclid's classic proof of the infinitude of primes today, the following question appeared to me. Let $p_1,p_2,p_3,\ldots$ be the prime numbers, listed in increasing order. Set $$k_n = ...
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0answers
30 views

The Mahler measure of a binary form and the natural action of a matrix ring

Let $F(x,y) = a_0 \prod_{i=1}^d(x-\alpha_iy)$ be a binary form of degree $d \geq 2$ and nonzero discriminant $D(F)$. Define the Mahler measure $M(F)$ of $F$ by $$M(F) = \left|a_0\right| \prod\limits_{...
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52 views

An inequality involving the Mahler measure and the discriminant of a polynomial

Let $F(x,y) = a_0 \prod_{i=1}^d(x-\alpha_iy)$ be a binary form of degree $d \geq 2$ and nonzero discriminant $D(F)$. Define the Mahler measure $M(F)$ of $F$ by $$M(F) = \left|a_0\right| \prod\limits_{...
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0answers
60 views

Popescu-Neron Desingularization for K[[T_1,…,T_∞]]

Let $K[[T_1,...,T_n]]$ be a finitely many variables formal power series ring over a field $K$. Dorin Popescu proved that there are smooth algebras $A_{\lambda}$'s which are of finite type over $K$ ...
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2answers
523 views

Why is 12 the smallest weight for which a cusp forms exists

I have already asked this question on MathStackExchange (here) but did not receive an answer, therefore I am trying it here: On wikipedia (here) I have read the following: Twelve is the smallest ...
3
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2answers
149 views

$p$-simple integers from between $n$ and $n+p-1$

Let $\ p\ $ be an arbitrary prime. Then an integer $\ s\ $ is called $p$-simple $\ \Leftarrow:\Rightarrow\ s\ $ is not divisible by any prime $\ q<p.\ $ Could you prove my conjecture (or is it ...
11
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1answer
201 views

A Galois extension over $\mathbb{Q}$ with Galois group $A_4$ and with cyclic decomposition groups

Does there exist a Galois extension $L/\mathbb{Q}$ with Galois group $A_4$ (the alternating group on four letters) such that all the decomposition groups are cyclic? This question is motivated by the ...
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0answers
260 views

Is the number $\sum_{p\text{ prime}}p^{-2}$ known to be irrational?

Is the number $$\sum_{p\text{ prime}}p^{-2}$$ known to be irrational? The limit exists, since $$\sum_{p\text{ prime}}p^{-2}<\sum_{i=1}^{\infty}i^{-2}=\frac{\pi^{2}}{6}$$.
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52 views

Coherence of subrings of K[[X,Y]]

Let $K[[X,Y]]$ be a two-variables formal power series ring over a field $K$. Consider a sub-ring $\iota \colon A \subset K[[X,Y]]$. Q. Is A coherent? $\quad$ Or is it automatic that $\iota$ is ...
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55 views

Congruent numbers and primorials

The first 10 primorials (2, 6, ... , 6469693230) are congruent numbers subject to the Birch Swinnerton-Dyer conjecture. My question is - What is the first primorial not to be a congruent number (...
5
votes
1answer
603 views

Ordinary primes vs supersingular primes

Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM. As shown by Serre, the set of supersingular primes for $E$ has density zero. Is the analytic rank of $L(E,1)$ determined only by the ...
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2answers
137 views

A multidimensional version of Hensel's lemma? (for more than one polynomial)

The classical Hensel's lemma is stated as follows: Let $f(x) \in \mathbb{Z}_p[x]$ and $a \in \mathbb{Z}_p$ satisfy $$ |f(a)|_p < | f'(a) |_p^2. $$ Then there is a unique $\alpha \in \mathbb{Z}_p$...
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0answers
42 views

Sieving for points not on a conic and quadratic residues

Let $$\displaystyle F(x,y,z) = \sum_{\substack{0 \leq i_1, i_2, i_3 \leq 2 \\ i_1 + i_2 + i_3 = 2}} a_{i_1, i_2, i_3} x_1^{i_1} x_2^{i_2} x_3^{i_3}$$ be a ternary quadratic form with integer ...
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1answer
98 views

Quadratic (and otherwise) squares, part II

This is a follow-up to this question: Quadratic squares Consider now a polynomial in two variables $f(x, y).$ Are there bounds (upper or lower) for how many $x_0$ of height less than $N$ such that $f(...
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1answer
200 views

Can we find primitive Pythagoras triplet (in integers), with two sides being as powerful numbers?

Actually this seemingly elementary question had very much surprised me, I though it is quite easy to find such triples described in question, but strangely after trying so many times, couldn't find ...
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36 views

Representing sparse set as set of extremely good approximation

For $\alpha \in \mathbb R$ and $\varepsilon(n) > 0$, consider the set $$ N(\alpha,\varepsilon) = \{ n \in \mathbb{N} \ : \ \lVert \alpha n \rVert < \varepsilon(n) \} $$ (where $ \lVert x \rVert$...