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

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17 views

Can Shor's Algorithm be modified to run efficiently on a classical computer?

Shor's algorithm is an algorithm which factors integers in polynomial time on a quantum computer. If one tries to run it on a classical computer, one runs into the problem that the state vector that ...
4
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0answers
67 views

Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system, $$x_1^2+x_2^2+\dots+x_n^2 = y^2$$ $$x_1^3+x_2^3+\dots+x_n^3 = z^3$$ and define the function, $$F(s_m) = x_1+x_2+\dots+x_n$$ For $n\geq3$, using an ...
0
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0answers
23 views

Density of polynomials which are soluble with respect to a set of primes

Suppose that $p$ is a prime, and $f(x)$ is a polynomial with integer coefficients and positive degree. Then there exists an integer $n_p$ such that $p | f(n_p)$ if and only if $f(x)$ has a linear ...
0
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1answer
91 views

Are reduced residue systems relative primorials an active area of research? If not, why not?

As a math amateur, I am finding the study reduced residue systems relative a primorial a very interesting way to understand the distribution of primes. For example, it is fascinating to me that it is ...
1
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1answer
50 views

Counting primes of the form $p^{\frac{k}{t}} \left( t \in \mathbb{R}{+}, k \in \mathbb{N} \right)$ by changing $\rho$'s in $\psi(x)$?

With $\rho=\beta+ \gamma \,i$ being a non-trivial zero of $\zeta(s)$, the logarithmic prime counting function is: $$\psi(x) = x - \log(2\pi) - \frac12 \log\left(1- \frac{1}{x^2}\right) - ...
1
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1answer
89 views

Eisenstein series of weight $2$ for $\Gamma_0(N)$ : where am I wrong?

Let $A_{N,2}$ be the set of triples $(\psi,\varphi,t)$ such that $\psi$ and $\varphi$ are primitive Dirichlet characters modulo $u$ and $v$ with $(\psi\varphi)(-1)=1$, and $t$ is an integer such ...
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119 views

A question on (odd) perfect numbers

I have asked this question in MSE a few weeks back, but did not receive any responses. I have cross-posted it to MO, hoping that it is appropriate for this site. Let $\sigma(x)$ be the (classical) ...
2
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1answer
86 views

Density of polynomials with a prescribed number field extension

For any polynomial $f(x) \in \mathbb{Z}[x]$, let $K_f$ denote the minimum splitting field over $\mathbb{Q}$ which contains all of the roots of $f$. Let $n \geq 2$ be a fixed integer, and let $K$ be a ...
7
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2answers
145 views

approximate two different real numbers to order $\frac{1}{z^{3/2}}$

I took this result from Minkowski's book on Geometry of numbers: Two arbitrary real quantitites $a$ and $b$ may be made to approach as near as we wish in value the two fractions $\frac{x}{z}$ and ...
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43 views

Titchmarsh S function [on hold]

SO it is known that Titchmarsh S function $$ S(T)= \pi^{-1} arg\quad \zeta\bigg(\frac{1}{2}+iT\bigg)$$ under the assumption of riemann-hypothesis gives $$ S(T)=O(\frac{\log T}{\log \log T})$$ can ...
7
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1answer
204 views

What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...
1
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0answers
82 views

Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism

MOTIVATION Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...
5
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1answer
102 views

Bounding the number of lattice points inside an $n$-dimensional ellipsoid

I am wondering if it is possible to produce an upper and/or lower bound on the number of integer lattice points that lie inside an $n$-dimensional ellipse. That is, given an $n$-dimensional ellipsoid ...
3
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4answers
184 views

Uniform upper bound for the sum over primes $\sum_{p \leq x} p^{-1+\varepsilon}$

I am reading the article D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991) (click). In equation (27) the authors, apparently, used the following ...
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0answers
22 views

Degree $2$ nilpotent matrices with non-zero product [migrated]

Let $n$ be sufficiently large positive integer. Let $M_i$ be $n$ matrices with entries in either $\mathbb{C}$ or $\mathbb{Z}/n \mathbb{Z}$. Is it possible, (1) for $1 \le i \le n, M_i^2=0$ and ...
5
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0answers
138 views

Size of smallest multiple $\sum \epsilon_i p^i$ of $n$ with digits $\epsilon_i\in \{0,1\}$ for $p$ a prime

Any natural number $n$ coprime with a prime number $p$ is a divisor of $M=1+p+p^2+\dots+P^{N-1}$ where $N\leq \varphi((p-1)n)$ is the order of $p$ in $\left(\mathbb Z/((p-1)n)\mathbb Z\right)^*$. ...
10
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1answer
346 views

Class field towers

It is known (Golod and Shafarevich) that the class field tower of a finite extension $K$ of $\mathbb{Q}$ may be infinite. But is it always finite for $K=\mathbb{Q}[\zeta]$ where $\zeta$ is a root of ...
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0answers
77 views

X^2 + 4 = y^5 is there any solution to this [on hold]

is there any integer solution to the given equation? i have tried to solve the problem considering modulo of different prime but it just aint working it'd be nice if i get a solution soon
7
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1answer
185 views

Asymptotic limit of truncated Legendre sieve

Consider the truncated sum $$ S(x):=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}\mu(d)/d, $$ where $P(z)$ is the product of all primes less than or equal to $z$, and $\mu(d)$ is the Möbius ...
4
votes
1answer
260 views

Finite field “contour” sum

Let $\Bbb{F}_q$ be a finite field. Choose a non-square $\delta \in \Bbb{F}_q^*$ and form the quadratic extension $\Bbb{F}_q\big( \sqrt{\delta} \, \big)$. For an element $z \in \Bbb{F}_q\big( ...
3
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0answers
278 views

Explicit Galois Action for $X^3 - X -1$ [migrated]

I have always been frustrated with how indirect discussions of Galois Theory are in Algebra textbooks. Even in fine treatments such as Miles Reid. Are there any good examples where we can draw ...
7
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1answer
314 views

Is this theorem on $L$-functions known?

Notations For $f$ a meromorphic function on a domain $\Omega\subseteq \textbf{C}$, we shall say for convenience that $f$ is represented by an Ordinary Dirichlet Series (ODS) if $f$ can be written ...
17
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3answers
1k views

A Polynomial With Positive Prime Density

Let $P(x)$ be a non-constant polynomial with real coefficients. Can natural density of $$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$ be positive?
2
votes
0answers
130 views

Squarefree part of a Mersenne number

Consider the Mersenne number; $M_p=2^p−1$. Let $M_p=a_pb^2_p$ where $a_p$ is positive, squarefree, and $p$ is prime. A chinese paper written by Le Maohua "“On Mersenne Numbers”" states that the ...
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0answers
109 views

Groupes fondamentaux de Tate mixte [on hold]

Have anyone ever read deligne and Goncharov's paper,Could you give me some idea why the formula(5.16.1) is true?this paper is very easy to find on the internet.So please forgive me not giving the ...
6
votes
3answers
248 views

Nontrivial solutions for $\sum x_i = \sum x_i^3 = 0$

For $x_i \in \mathbb{Z}$, let $\{x_i\}$ be a fundamental solution to the equations: $$ \sum_{i= 1}^N x_i = \sum_{i=1}^N x_i^3 = 0 $$ if $x \in \{x_i\} \Rightarrow -x \notin \{x_i\}$. For instance, a ...
7
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0answers
165 views

Integer solutions of $x^2=4+8y^2+13z^2$

I have been looking for integer solutions of certain Diophantine equations, one of the simplest examples being $x^2=4+8y^2+13z^2$. The ideal answer would be a way to parametrize all the integer ...
3
votes
1answer
107 views

Atkin-Lehner theory for nonholomorphic Eisenstein series

I am currently reading something about nonholomorphic Eisenstein series $E_\mathfrak{a}(z,1/2+it)$ for $\Gamma_0(q)$, where $\mathfrak{a}$ is a cusp (cf. Iwaniec, H. Spectral Methods of Automorphic ...
15
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1answer
808 views

Evaluating a remarkable term for primes p = 5 (mod. 8)

Let $p > 3$ be a prime number, and $\zeta$ be a primitive $p$-th root of unity. I am interested in knowing the exact value of $$w_p = \prod_{a \in (\mathbb F_p^{\times})^2}(1 + \zeta^a) + \prod_{b ...
3
votes
1answer
150 views

Hodge-Tate weights of induced representation

Let $K$ be a finite extension of the field of $p$-adic numbers $\mathbb{Q}_p$ and let $E$ be another such extension, such that all the $\mathbb{Q}_p$ embeddings $K \to \bar{\mathbb{Q}}_p$ are ...
6
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1answer
276 views

Primes isolated by large gaps to either side

Say that the $n$-th prime $p_n$ is isolated to degree $k$ (my notation) if the prime gap to either side is larger than $\log p_n$ to the $k$-th power: \begin{eqnarray*} p_n - p_{n-1} & > & ...
7
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1answer
278 views

What's the difference between Euler systems and Kolyvagin systems?

Is there a difference between Euler systems and Kolyvagin systems - or do they refer to the same thing? For example there is the Heegner point Euler system, but you don't really see a Heegner point ...
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0answers
64 views

Integer solution to the equation [migrated]

Does there exists an integer solution (for every integer $m\geq 1$) for the following equation? $$x_1x_2...x_n+(2y+1)z+y=4m+3$$ where, $1\leq x_1\leq x_2\leq...\leq x_n\leq l$,$0\leq y \leq ...
0
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1answer
284 views

Reference for a lemma on étale maps

The Stacks Project has the following really nice Lemma concerning étale maps of rings: Let $A\rightarrow B$ be a finitely presented, étale morphism of rings. Then there exists a presentation $$ ...
3
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0answers
48 views

Noncommutivity of various lifts

Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ ramified at $p$ and $\infty$ for simplicity. Then an automorphic form on $D^{x}/F^{x}$ has several different interpretations. First, through ...
2
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2answers
325 views

Primes as uncorrelated random variables [closed]

The heuristic justification section of the Wikipedia article about Goldbach's conjecture says that the argument that suggests that the number of twin primes below $x$ should be roughly ...
3
votes
4answers
310 views

Examples of polynomials $x_1(t), x_2(t)$ such that $(x_1(t))^4 + (x_2(t))^4$ has a double root

Are there examples of polynomials $x_1(t), x_2(t) \in \mathbb{Q}[t]$ of equal degree at least one, with $\gcd(x_1(t), x_2(t)) = 1$, such that the sum $(x_1(t))^4 + (x_2(t))^4$ is divisible by the ...
5
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1answer
164 views

An upper bound for the length of the continued fraction expansion of $\sqrt d$

Let $d\ge 2$ and let $$ \sqrt d =[a_0; \overline{a_1,\dots, a_\ell, 2a_0}] $$ be its continued fraction expansion. Clearly, if $d=n^2+1$, then $\ell=0$, which gives the lower bound for $\ell$. ...
4
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1answer
140 views

Sets of natural numbers such that sums of a bounded number of its elements form a semigroup

This is a naive question and I'm afraid it might be better placed on math.se. I would like to leave it to your judgement. I would like to know what is known about sets $A$ of natural numbers such ...
47
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4answers
2k views

When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true. For example, it gives some evidence that there are finitely many ...
13
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1answer
438 views

When complex conjugation lies in the center of a Galois group

Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question ...
14
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3answers
2k views

A variant of Goldbach Conjecture

I'm asking if this variant of weak Goldbach's Conjecture is already known. Let $N$ be an odd number. Does there exists prime numbers $p_1$, $p_2$ and $p_3$ such that $p_1+p_2-p_3=N$? Ideally, can we ...
11
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2answers
454 views

distribution of $\sqrt{-1} \mod p$

While reading up on quadratic reciprocity, I learned that if $p = 4k+1$ then $-1$ has a square root in $\mathbb{Z} / p \mathbb{Z}$. Let $r_p$ be an integer with $0\leq r_p < p$ and $r_p^2 \equiv ...
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0answers
45 views

how to calculate the radius of convergence of the p-exponentials of Pulita?

please it is known from the Pulita thesis that the radius of convergence of his pi-exponentials is 1; he used a differential operator which has this pi-exponetial as a solution etc..., but me I ...
4
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2answers
590 views

Is every positive integer a sum of at most 4 distinct quarter-squares?

There appears to be no mention in OEIS: Quarter-squares, A002620. Can someone give a proof or reference? Examples: quarter-squares: ${0,1,2,4,6,9,12,16,20,25,30,36,...}$ 2-term sums: ${2+1, 4+1, ...
15
votes
1answer
531 views

Question on the irrationality of $e$

I was surprised that the numbers $\pi$, $\ln{(2)}$, $\zeta{(2)}$, and $\zeta{(3)}$ can be shown to be irrational in what seems to be "three-lined proofs" (as identified here on Overflow: Establishing ...
7
votes
1answer
210 views

how do automorphisms of elliptic curves act on the Tate module?

Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T_\ell(E)$ ($\ell\ne p$) with ...
5
votes
1answer
136 views

On one class of Somos-like sequences

This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer? Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence ...
6
votes
0answers
97 views

Integers $d$ for which the Negative Pell equation is soluble for both $d$ and $2d$?

Let $\text{NPE}_d$ denote the negative Pell equation: $$ x^2-dy^2=-1$$ Where $d$ is a given positive nonsquare integer and integer solutions are sought for x and y. we know that (in this paper): ...
1
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0answers
257 views

Generalization of proposition of Granville related to abc conjecture

Related to this question. For polynomial $f$, let $rad(f)$ denote the radical of $f$, the product of irreducible factors. Suppose that $G(x,y) \in \mathbb{C}[x,y]$ is homogeneous without any ...