# Tagged Questions

**4**

votes

**0**answers

117 views

### Primes and Parity

This problem is motivated by the polymath4 project. There, the aim was to find an efficient deterministic algorithm for finding a prime larger than $N$. The hope was to find a polynomial algorithm in ...

**12**

votes

**2**answers

328 views

### A possibly surprising appearance of Lucas numbers

Let $S$ be the set of polynomials defined as follows: $0$ is in $S$, and if $p$ is in $S$, then $p + 1$ is in $S$ and $x \cdot p$ is in $S$, so that $S$ "grows" in generations: $g(0)=\{0\}$, ...

**1**

vote

**0**answers

51 views

### A question related to metric Diophantine approximation

In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that
$$
\left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q}
$$
has ...

**0**

votes

**0**answers

178 views

### Constructing a particular Cayley Graph

Pick a positive number $P$.
Pick $M$ positive numbers $g_1<\dots<g_M$ each less than $P-1$.
Denote $\mathcal{S}=g_1,\dots,g_M$.
Denote $G=G_{P}[\mathcal{S}]$ to be Cayley graph (considered ...

**4**

votes

**0**answers

277 views

### Survey of Erdős' “Tricks” [on hold]

Is there a kernel of "tricks", techniques and tools that Paul Erdős was particularly fond of and therefore employed a lot in his research work? Could you point out some survey papers that deal with ...

**7**

votes

**1**answer

118 views

### Orders in Central Simple Algebras. Applications

It is known that orders in quaternion algebras (over a number field) are used for constructing geometric objects like hyperbolic orbifolds and Shimura curves. Moreover, if one knows embedding ...

**3**

votes

**0**answers

207 views

### Is there a hidden symmetry in the prime numbers distribution?

Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime.
...

**2**

votes

**0**answers

86 views

### Number of common solutions of polynomial system

Let $ \mathbb{F}_p$ be a finite field and $\{f_j\}_{j=1}^{j=r} \subseteq \mathbb{F}_p[X_1,...,X_n]$ be a set of polynomials.
Let consider the system of equations:
$f_j(x_1,...,x_n)=0$ for $j = ...

**20**

votes

**4**answers

675 views

### Parametric solutions of Pell's equation

Given a positive integer $n$ which is not a perfect square, it is well-known that
Pell's equation $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$.
Question: Let $n$ be a ...

**7**

votes

**1**answer

141 views

### A positivity problem involving the number of ways of expressing $n$ as a product of $k$ factors

Let $d_k(n)$ denote the number of ways of expressing $n$ as a product of $k$ factors, and let $$D_k(x)=\sum_{n\leq x}d_k(n)$$ be the summatory function. During a study of Mertens' function I was lead ...

**0**

votes

**0**answers

89 views

### Coefficients of Hilbert polynomials

Recall that we can define the Hilbert series of a graded commutative algebra
$$\displaystyle S = \bigoplus_{n \geq 0} S_n$$
over a field $K$ by
$$\displaystyle \mathcal{H}_S(t) = \sum_{n=0}^\infty ...

**0**

votes

**0**answers

43 views

### About the selection of reals $u_0,u_1$ such that $u_{n}$ is a positive integer

Let $r\geq 4$ and $n≥1$ be two positive integers. Let us consider the sequence $(u_{n})$ defined by:
$$u_{n}=r^{n^2}\sum_{m=1}^{n}\frac{u_1-ru_0+2(m-1)}{r^{m^2}}+u_0$$ where $u_0,u_1$ are real ...

**3**

votes

**1**answer

169 views

### References for general Hasse-Weil zeta function

Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case.
I am ...

**2**

votes

**0**answers

89 views

### Comparing the size of two sums

Let $q$ be a prime power and $\mathbb{F}_q$ be the field of $q$ elements.
Let $\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$.
I am working on a research project, where I bounded a ...

**2**

votes

**0**answers

66 views

### Asymptotic expansion for the Bell numbers

The Bell numbers $B(n)$ (that is, the numbers that count the set partitions of a set, and have exponential generating function $\exp(e^x -1)$ ) admit the asymptotic expansion
$$\frac{\log B(n)}{n} = ...

**2**

votes

**1**answer

142 views

### Rankin-Selberg convolution and product of degrees

As I'm kinda obsessed with the Selberg class and because of the general converse conjecture, I'm still trying to get a rough idea of what automorphic representations and their L-functions as well as ...

**2**

votes

**1**answer

345 views

+50

### Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from $1$

Found this on Complexity Zoo warning expired certificate
check NP Over The Complex Numbers.
[BCS+97] show the following striking result. For a positive integer $n$, let $t(n)$ denote the minimum ...

**12**

votes

**1**answer

516 views

### Elliptic curves and connected components

Are there elliptic curves of positive rank with two real connected components
in which all the rational points lie only on one component?
Concrete examples are really appreciated.

**-5**

votes

**0**answers

132 views

### perfect numbers and their properties [closed]

Yesterday I asked a question about perfect numbers. After thinking about the answer and comments I received, I now conjecture that the cube of any perfect number can be written in the form of the sum ...

**18**

votes

**1**answer

603 views

### What are the products $\prod_{A\subset{\mathbb F}_p\colon |A|=n} \sum_{a\in A} \zeta^a$ equal to?

This is a somewhat more explicit version of a question I have recently asked.
Let $p$ be an odd prime, and write $\zeta:=\exp(2\pi i/p)$ (any other primitive $p$th root of unity will do as well). For ...

**8**

votes

**1**answer

369 views

### Open access journals in number theory

I'm a phd student in number theory, and I'm required (by the funding council) to publish any article I write in open access journals. The problem is that all journals I can find are either out of my ...

**4**

votes

**1**answer

127 views

### Length of the binary representation of a primorial

Let $p_n\# = \prod_{k=1}^n p_k$ be the $n$-th primorial.
Q1. Given $n$ (in binary) is there an efficient way (polynomial time) to calculate the exact number of digits of the binary representation ...

**2**

votes

**2**answers

536 views

### Has this formula about prime gaps already been conjectured and/or proven?

While playing around with prime gaps, I found out that the following formula seems to be a rather good approximation of the ratio $\dfrac{p_{b}-p_{a}}{b-a}$ where $a<b$ are positive integers:
...

**8**

votes

**0**answers

214 views

### Transcendence of products of certain real algebraic numbers

Let
\begin{equation}
z := \prod_p p^{1/p^2},
\end{equation}
where the product is over all prime numbers $p$, and we always take the positive real root. Is $z$ transcendental or algebraic, or (as I ...

**0**

votes

**1**answer

101 views

### Fixed field of the Nebentypus of a newform for $\Gamma_1(N)$

Let $f=\sum_{n\geq 1}\in S_2(\Gamma_1(N),\varepsilon)$ be a normalized newform without CM and with Nebentypus $\varepsilon$. Let $L=\mathbb Q(a_n\colon n\in \mathbb N)$ be the number field generated ...

**-8**

votes

**1**answer

160 views

### Proof of a cubic equation problem [closed]

Well I was doing some questions and i found something. This equation
$x^3+y^3+z^3=w^3$
has only one solution which is
$x=3,y=4,z=5,w=6$.
And what I have have proposed is that there is not other ...

**3**

votes

**1**answer

100 views

### Density of tuples of conjugate algebraic numbers

One can see that algebraic numbers are dense in the complex plane by just looking at quadratic polynomials. I am interested in a "higher order" density of algebraic numbers.
More specifically: is it ...

**6**

votes

**1**answer

394 views

### An old paper of S.Chowla on unit equations

It is referenced that in
Chowla, S., Proof of a conjecture of Julia Robinson, Norske Vid. Selsk. Forh. (Trondheim) 34, 100–101 (1961),
it is shown that the equation $\epsilon_1 + \epsilon_2 = 1$ ...

**8**

votes

**2**answers

310 views

### Quintic polynomials generating cyclic extensions

We know that a cubic equation generates a cubic cyclic extension iff it has a perfect square discriminant. Now I am wondering if there is a similar condition for quintic polynomials. So I am trying to ...

**6**

votes

**1**answer

132 views

### Mean value of Maass forms

Let $X = SL_2(\mathbb{Z}) \backslash \mathbb{H}$ be the modular surface. Consider a basis of $L^2$-normalized Hecke-Maass cusps forms $\phi_j$ on $X$ with $-\Delta$-eigenvalue $\lambda_j$. ...

**2**

votes

**0**answers

103 views

### Thin sequences with good counting properties

I am looking for sequences $\{a_n\} \subset \mathbb{N}$ with the following properties:
(1) $\displaystyle \# \{a_n \leq x \} = \frac{x}{\sqrt{\log x}} + O_\epsilon(x^{1/2 + \epsilon})$, and
(2) $\# ...

**7**

votes

**1**answer

187 views

### higher reciprocity theorems from ratios of Gauss sums

One recent proof of quadratic reciprocity involves computing various rations of the Gauss sum.
In Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation Gurevich, ...

**0**

votes

**0**answers

43 views

### Estimating solutions to a binary form congruence with small moduli and prime inputs

Currently I am dealing with the following problem. Suppose that $F(x,y) \in \mathbb{Z}[x,y]$ is a binary form of degree $D \geq 2$ and $k \geq 2$ is an integer such that for all primes $p$, there ...

**1**

vote

**0**answers

408 views

### Is $ a^2b^2-ab-ap$ a perfect square for suitable $a,b\in \mathbb{Z}^+$?

Consider the expression
$$
a^2b^2-ab-ap\qquad (a,b\in \mathbb{Z}^+),
$$
where $p\equiv1\pmod{4}$.
Question. For every prime $p\equiv1\pmod{4}$ do there exist $a,b\in\mathbb{Z}^+$ such that ...

**8**

votes

**0**answers

142 views

### Do all complex zeros in the strip of $\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$ lie on the critical line?

Numerical evidence suggests that the complex zeros of:
$$f(s):=\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$$
all reside on the line $\Re(s)=\frac12$, except for a finite few outside ...

**1**

vote

**0**answers

373 views

### Can one generate a sequence of natural numbers whose density has a given distribution? [migrated]

Suppose $\{ p_{k} \}$ is a collection of real numbers with the following
properties:
1) $p_k \in (0,1)$ $~~~~$(i.e. $0$ and $1$ are not allowed values)
2) $\sum_{k=1}^{\infty} p_k =1$
An ...

**22**

votes

**0**answers

598 views

### On certain representations of algebraic numbers in terms of trigonometric functions

Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ...

**4**

votes

**0**answers

183 views

### A relation between the Gamma function and the Mobius function?

It is well known how altering the integral for the Gamma function:
$$\displaystyle \Gamma(s) = \int_0^\infty t^{s-1} e^{-t}\,dt$$
through substituting $t=nx$,
$$\displaystyle \Gamma(s)\frac{1}{n^s} ...

**2**

votes

**1**answer

226 views

### Prime Number Theorem on APs under various conjectures

I'm trying to find the best asymptotic expansions for $\pi(x; a, q)$ in various states:
Unconditionally we have
\begin{equation}
\pi(x; a, q) = \frac{\operatorname{li(x)}}{\phi(q)} + O\left(x ...

**6**

votes

**1**answer

336 views

### Some questions about the ring Z((x))

$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\dim}{\text{dim }}$
Let me begin by apologizing for the length of this question, but I thought this might be interesting to some of you. This ring isn't ...

**5**

votes

**0**answers

392 views

### For which rational values of $c$ and $d$ are the numbers $\sin{(\pi\cdot c)}$, $\sin{(\pi\cdot d)}$ and $1$ linearly dependent over $\mathbb{Q}$?

A year ago, I posted this problem on [MSE]. After a number of edits, I have arrived at the following more general problem (suggested by Hjalmar Rosengren; see the comments below).
For which ...

**33**

votes

**2**answers

1k views

### Why does this sequence converges to $\pi$?

One of my daughters was having a small programming exercise.
Let's consider following algorithm:
Take a list of length $n$: $\ (1\,\ 2\,\ \ldots\,\ n)$.
Remove every $2$nd number.
From the ...

**13**

votes

**3**answers

2k views

### How did Ramanujan discover this identity?

Let $$\small F_n=(a+b+c)^n+(b+c+d)^n-(c+d+a)^n-(d+a+b)^n+(a-d)^n-(b-c)^n$$ and
$ad=bc$, then
$$64*F_6*F_{10}=45*F_8^2$$
This fascinating identity is due to Ramanujan and can be found in ...

**4**

votes

**1**answer

496 views

### The elliptic curve for $x_1^9+x_2^9+\dots+x_6^9 = y_1^9+y_2^9+\dots+y_6^9$

I. If there are $a,b,c,d,e,f$ such that,
$$a+b+c = d+e+f\tag1$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag2$$
$$3u^3-3uv+w=-def\tag3$$
where $u=a+b+c,\; v = ab+ac+bc,\;w = abc$, then,
$$(a + u)^k + (b + ...

**2**

votes

**0**answers

77 views

### Buildings associated to generalized $BN$ pairs

I'll begin by asking a general question, and then specializing to the situation I really care about.
Let $G$ be a group and let $(B, N)$ be a $BN$-pair in $G$ (see, for instance, page 39 of Tits' ...

**0**

votes

**0**answers

129 views

### Does the proportion of non trivial zeroes of given real part increase on [0,1/2] for all L-functions?

Here what I call an L-function is either an element of the Selberg class or an automorphic L-function. For such a function $F$, $x\in [0,1/2]$ and $T\gt 0$, let's define $\delta_{F,T}(x)$ such that ...

**4**

votes

**0**answers

136 views

### Automorphisms of k((X))

I'm looking for a good reference for the following fact:
Let $k$ be a perfect field of characteristic $p$ and let $K=k((X))$.
Then every $k$-linear automorphism of $K$ is continuous with respect
to ...

**2**

votes

**1**answer

877 views

### Why Riemann Hypothesis so important [closed]

I am often hearing people emphasized how important the RH is,one of them said that it should lead to an efficient way of determining whether a given large number is prime,and the other said,RH would ...

**5**

votes

**1**answer

297 views

### An old conjecture of M.Newman

M.Newman raised several questions in his 1957 paper on modular forms.
Definition: $H_n$ is the subclass of all zero-free weakly modular forms of weight 0 on $\Gamma_0(n)$, where $n$ is a composite ...

**3**

votes

**0**answers

119 views

### The Bilu-Linial conjecture and Ramanujan graphs

The Bilu-Linial conjecture claims that every $d-$regular graph has a $2-$lift such that for the signing matrix has its eigenvalues between $[-2\sqrt{d-1},2\sqrt{d-1}]$ (the ``signing matrix" is the ...