# Tagged Questions

**0**

votes

**0**answers

3 views

### Langlands program vs Shimura-Tanayama-Weil conjecture

Edward Frenkl said that "we can see Langlands program as a generalization of Shimura-Tanayama-Weil conjecture in the case of elliptic curves "
I hope I'm not distorting his phrase, can someone ...

**63**

votes

**0**answers

4k views

### Grothendieck-Teichmuller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmuller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here ...

**-3**

votes

**0**answers

135 views

### Can the work of Hardy & Ramanujan about partitions shed light on Hardy-Littlewood's k-tuple conjecture? [on hold]

If I'm not mistaken, Hardy and Ramanujan produced an asymptotic formula for the number of partitions of an integer that was later shown to be an exact formula by Selberg.
But Hardy also formulated ...

**-5**

votes

**0**answers

47 views

### I've read on the internet that 43 is congruent to 8 modulo 5+3w, can you explain? [on hold]

Initially, I wanted to determine whether 19 is a cube modulo 43.
Reading online, I've come across this computation that includes
a step that I don't understand. The step in question states that
(43 ...

**1**

vote

**0**answers

120 views

### What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [on hold]

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$.
If $L$ is a number field ...

**5**

votes

**0**answers

184 views

### Average size of $p$-part of the Tate-Shafarevich group for elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve defined over $\mathbb{Q}$. For a given prime $p$, the $p$-Selmer group $\operatorname{Sel}_p(E)$ of $E$ and the $p$-part of the Tate-Shafarevich $Ш_E[p]$ group ...

**2**

votes

**1**answer

204 views

### Sampling from random totally unimodular matrices of a particular type?

Is there a way to parametrize totally unimodular $(3n+2)\times(2n+2)$ matrices of form
$$\begin{bmatrix}
\pm1 & \pm1 & 0 & 0 &\dots & 0 & 0 & 0 & 0\\
A_{2n} & ...

**0**

votes

**0**answers

82 views

### $z^n-t^m=x^3+y^3$ and Vojta's more general abc conjecture

In A more general abc conjecture, p. 7 Paul Vojta conjectures:
If $x_0,\ldots x_{n-1}$ are nonzero coprime integers satisfing $x_0 + \cdots x_{n-1}=0$
$$ \max\{|x_0|,\ldots |x_{n-1}|\} \le C ...

**1**

vote

**3**answers

501 views

### Consecutive composite numbers

Let $C(N,k)$ be the smallest positive integer $x$ such that $[1,x]\subset \mathbb{Z}$ contains $k$ disjoint intervals $I_1, ..., I_k$ of $N$ consecutive integers that are all composite. (For example, ...

**24**

votes

**5**answers

2k 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 ...

**3**

votes

**1**answer

179 views

### Self-containing trees

Suppose that $r^2-r-1=0$ and that $T$ is the tree with root $1$ such that the children of each node $x$ are $rx$ and $x+1$. Remove all duplicates as they occur, and let $T(r)$ denote the remaining ...

**18**

votes

**3**answers

391 views

### Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...

**1**

vote

**1**answer

84 views

### The exponential Diophantine equation $a^n-b^m=x^3+y^3$ for arbitrary large $n,m$

Let $a,b$ be coprime integers, neither a perfect power, $n,m$ naturals and $x,y$ integers.
Consider the exponential Diophantine equation
$$ a^n-b^m=x^3+y^3 \qquad (*) $$
Nontrivial solution ...

**8**

votes

**1**answer

246 views

### Have the explicit Poisson-type formulas of Guinand and Meyer been observed before?

In a recent paper of Meyer Measures with locally finite support and spectrum PNAS vol. 113 no. 12:3152–3158 (behind a paywall, but see also these seminar notes) some new explicit Poisson-type formulas ...

**6**

votes

**2**answers

483 views

### How many integers are of the form $n/d(n)$, where $d(n)$ is the number of divisors of $n$?

This was post by me on Maths SE: but it did not get any solution.
Some months ago I made the following conjecture -
Let $d(n)$ denote the number of divisors of $n $.
Then let $N$ be a number such ...

**1**

vote

**0**answers

166 views

### Generalized arithmetic progressions contained in Bohr sets

Recall that a generalized arithmetic progression of dimension $d$ is, by definition, a set of the form $P = P_1+\dots+P_d$, where $P_j = \{lp_j\ \mid \ |l|\leq l_j\}\subset \mathbb Z$ is an ordinary ...

**3**

votes

**1**answer

125 views

### 4-th order diophantine equation

I met in many places the equation
$(a^4-b^4)(c^4-d^4)=\square$
It is well known that this was investigated by Euler.
But I was unable to find the general solution of this equation. Could you please ...

**2**

votes

**6**answers

619 views

### Isotropic ternary forms

It is well known that some questions about isotropic ternary forms reduces to the study of the special case $f_0(X)=xz-y^2, X=(x,y,z)$, see page 301 of Cassel's "Rational quadratic forms" (Dover, ...

**-1**

votes

**0**answers

44 views

### General Solution of Quadratic Diophantine Equation System

I am looking for the general solution of:
$x^2-y^2=\square$
$x^2-z^2=\square$
$y^2-z^2=\square$
I have found a solution in tpiezas site. But not really sure if this is the general solution.

**0**

votes

**0**answers

26 views

### Is it possible to have an even superperfect number and an odd superperfect number whose product is an almost perfect number?

A number $n \in \mathbb{N}$ is said to be superperfect if
$$\sigma(\sigma(n)) = 2n.$$
A number $m \in \mathbb{N}$ is said to be almost perfect if $$\sigma(m) = 2m - 1.$$
Here is my question:
Is ...

**1**

vote

**1**answer

82 views

### Formula for negative polylogarithms

Theorem. We have that $\displaystyle \underbrace{x\left(\dfrac{d}{dx}\left(\cdots x \left(\dfrac{d}{dx} \left( \dfrac{x}{1-x}\right)\right)\cdots\right)\right)}_{\text{$x \frac{d}{dx}$ $m$ ...

**2**

votes

**1**answer

149 views

### Find a subset such that its sum is divisible by $n$

It is said that the following proposition is true.
$\forall S \subset \mathbb{Z}, |S| = 2n-1.\ \exists A \subset S, |A| = n$ which satisfies
$$
n \ | \ \sum_{a \in A}a.
$$
Could someone gives a ...

**3**

votes

**1**answer

147 views

### Do the roots of this equation involving two Euler products all reside on the critical line?

This question loosely builds on the second part of this one.
Take the Riemann $\xi$-function: $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. Numerical ...

**0**

votes

**0**answers

102 views

### More generalized RSA construction

Is there a way to construct RSA type cryptosystem over general number rings?
Can Number Field Sieve technique be applied here?

**2**

votes

**1**answer

132 views

### Is there a bijection $f: N \times N \rightarrow U \subset N$ with $f(x,y)+f(u,v)=f(x+u,y+v)$ and $f(x,y) \cdot f(u,v)=f(x \cdot u, y \cdot v)$?

Is there a subset of natural numbers that has the same additive and multiplicative structure as the set of ordered pairs of natural numbers under the classical operations of addition and ...

**14**

votes

**3**answers

4k views

### Are the nontrivial zeros of the Riemann zeta simple?

A few years ago, I found on arXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann zeta function were all ...

**7**

votes

**1**answer

344 views

### Lattice points near a curve

Bombieri and Pila had a well known bound for the count of lattice points on an algebraic curve in the plane. Does it generalize to a bound for the count of lattice points near (say within a distance ...

**0**

votes

**1**answer

301 views

### Q re Kaprekar's fixed mapping points

Jens Kruse Andersen in his comment in OEIS's A099009 noticed 3 families of numbers among Kaprekar's fixed mapping points (otherwise known as kernels of the Kaprekar's routine):
"Let $d(n)$ denote ...

**1**

vote

**2**answers

262 views

### Is it true that $\Phi_n(2)$ has a divisor of the form $kn+1$ for all $n\neq 6$?

Let $\Phi_n(x)$ be the $n$ th cyclotomic polynomial. I've checked the values of $\Phi_n(2)$ for some small $n\geq 2$ and noticed that there is always a divisor of $\Phi_n(2)$ of the form $kn+1$ ...

**2**

votes

**0**answers

101 views

### Modified Jacobi’s theta function

Be $t\in\mathbb{R}_0^+$.
Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$
Therefore $$\sum\limits_{k=1}^\infty ...

**2**

votes

**1**answer

258 views

### A quadrant of residues

Assume that following inequality holds $$\mathsf{w,x,y,z<AB,AC,AD,BC,BD,CD<ABC,ABD,ACD,BCD<wx,wy,wz,xy,xz,yz}$$ with $$\mathsf{gcd(A,B)=gcd(A,C)=gcd(A,D)=gcd(B,C)=gcd(B,D)=gcd(C,D)=1}$$
...

**17**

votes

**0**answers

337 views

+50

### Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space

Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all
$x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...

**8**

votes

**1**answer

181 views

### Arbitrarily many primes in a Fibonacci-type sequence

It is conjectured that the standard Fibonacci sequence contains infinitely many primes. While this is perhaps too difficult, I am wondering about the following simpler version:
Question. For any $K$, ...

**8**

votes

**0**answers

157 views

### Computing endomorphism rings of supersingular elliptic curves

I would like to know what algorithms there are to compute the linearly independent generators $(1,i,j,k)$ for quaternion algebra containing the endomorphism ring of a supersingular curve. The curve in ...

**3**

votes

**0**answers

343 views

### Finding the number of rational points effectively

Consider $\# P$ and $\oplus P$.
There is a $\# P$-hard problem: to find number of rational solutions of a system of polynomial equations over $\mathbb{F}_2$. The corresponding $\oplus P$-hard ...

**1**

vote

**1**answer

132 views

### On even almost perfect numbers other than powers of two

(Note: This question is an improved version of and has been cross-posted from this MSE post.)
Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an ...

**6**

votes

**2**answers

589 views

### Generalizing Ramanujan's “1729 story”

Whenever I read the anecdote about Hardy, Ramanujan and the taxi number 1729 I'm amazed that it could have occurred to anyone just off the top of their head that 1729 can be written as the sum of two ...

**0**

votes

**0**answers

189 views

### On the coherence of formal power series ring

Let $A = {\Bbb F}_p[[X_1,X_2,...]]$
be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$
$A$ consists of such formal sum elements as $\sum ...

**11**

votes

**2**answers

753 views

### Tate uniformization of nonsplit semistable elliptic curves

Let $E/\mathbf{Q}_p$ be an elliptic curve having split multiplicative reduction. Then Tate uniformization gives a surjective homomorphism of $p$-adic analytic groups $G_m \to E$, with infinite cyclic ...

**1**

vote

**0**answers

58 views

### For which primes $p$ is the field $\mathbb{Q}(\Gamma(1/p^{j}))$ a strict subfield of $\mathbb{Q}(\Gamma(1/p^{i}))$ whenever $0<i<j$?

I already asked this question on a French math forum but eventually came to think that as silly it may turn out to be, perhaps something interesting could finally emerge from it, so I decided to take ...

**4**

votes

**1**answer

235 views

### Does anyone recognize this exponential sum?

For $a$, $b$ two integers, let $(a,b)$ denotes their gcd. We define the following exponential sum :
$$G_q(n):=\sum_{d|q,~(d,q/d)=1}{e^{2i\pi n\frac{dd'}{q}}}$$
for $n$ a non-negative integer and $q$ ...

**0**

votes

**2**answers

137 views

### Function that gives 1 only when an integer is divisible by another integer [closed]

I need a function that takes two inputs, a and b, and returns 1 only when a is divisible by b and 0 otherwise. Can this be written in a nice mathematical way (other than using indicator functions)?

**3**

votes

**1**answer

405 views

### origin of analogy “primes as the atoms of number theory/ arithmetic”

a math student recently challenged me on the old comparison/ analogy of prime numbers to "the atoms of number theory or arithmetic" and then was wondering the origin of the phrase.
where does this ...

**0**

votes

**0**answers

110 views

### A number theory question [duplicate]

Let $m,n$ be integers and $P$ be a prime and $\lambda$ be an integer such that $0 < \lambda <\sqrt p$, Can the following equation have any answer for any $m,n,p,\lambda$ except $\lambda=+-1$?
...

**4**

votes

**1**answer

106 views

### Nearly Be Bruijn sequences constructed from De Bruijn sequences

Let $w$ be a De Bruijn $01$-sequence of the type $B(2,n)$; i.e., a cyclic $01$-sequence that contains every $n$-digit $01$-sequence exactly once. Let $x$ be a $01$-sequence of length $n$. When and ...

**0**

votes

**0**answers

87 views

### A number theory question related to algebraic graph theory? [closed]

Let $m,n$ be integers and $P$ be a prime and $\lambda$ be such that $0 < \lambda <\sqrt p$, Can the following equation have any answer for any $m,n,p,\lambda$?
...

**2**

votes

**1**answer

68 views

### On cluster points of a particular sequence

This is the sequel of a previous question.
Let us consider the sequence
$$
\xi_n = 2n \{n\xi\}-n,
$$
where $\xi>0$ is a given real irrational number and $\{\cdot\}$ is the fractional part.
Do ...

**-2**

votes

**2**answers

159 views

### Precise asymptotic of diophantine approximation

I know that every irrational number $\xi$ can be approximated by rational numbers in such a way that
$$
\left| \xi - \frac{p}{q} \right| \leq \frac{c}{q^2}
$$
for infinitely many choices of $p$ and ...

**5**

votes

**2**answers

510 views

### Is the Crandall, Dilcher and Pomerance heuristic concerning Wall-Sun-Sun primes still state of the art?

This is a question about the open problem Fibonacci divisibility from the Open Problem Garden.
The problem, originally stated in 1960 by D.D. Wall, has several equivalent formulations one of which ...

**2**

votes

**1**answer

182 views

### Number of rational points in a non-smooth variety

Let $X$ be an algebraic variety over $\mathbb{F}_q$ with dimensional $n$. We know that if $X$ is smooth than $X$ has about $q^{nk}$ rational points over $\mathbb{F}_{q^k}$ (Weil hypothesis). Is there ...