# Tagged Questions

**2**

votes

**0**answers

44 views

### Varieties with few monomials and the n-conjecture

The n-conjecture
is a generalization of abc and basically says that the if
$a_1 + \ldots + a_n=0$, no proper subsum vanishes and $a_i$
are coprime, then the radical of $a_1\cdots a_n$ can't be
too ...

**5**

votes

**1**answer

169 views

### p-adic Stein spaces

The higher cohomology of coherent sheaves vanish on Stein spaces (both complex and p-adic). In the case when the space ($X$) is a curve and we're working in the complex world, this shows that all ...

**1**

vote

**1**answer

74 views

### Does the Lehmer quintic parameterize certain minimal polynomials of the $p$th root of unity for infinitely many $p$?

The solvable Emma Lehmer quintic is given by,
$$F(y) = y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$
with discriminant $D = (7 + ...

**3**

votes

**0**answers

231 views

### Zeta function double product

Is it possible to write the following double product in terms of the zeta function?
\begin{align}
&\prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}}
\end{align}
Extending the ...

**7**

votes

**2**answers

295 views

### Least supersingular prime

Given an elliptic curve over the rationals, what can one say about the size of the smallest supersingular prime?

**11**

votes

**4**answers

461 views

### Zeros of the derivative of Riemann's $\xi$-function

The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real ...

**7**

votes

**2**answers

286 views

### Numbers with all N-digit prefixes divisible by N

In base 10, the number 3816547290 contains every digit exactly once. When I take the first N digits, that substring is divisible by N. For example, 381 is divisible by 3, 38165 is divisible by 5, etc. ...

**0**

votes

**1**answer

107 views

### Short arithmetic progressions in quadratic residues

Let p be a prime number of form 4k+1.
I guess that there are c(d) number of 3-terms arithmetic progressions (AP) in the set of quadratic residues modulo p, where c(d) is an integer constant depending ...

**-2**

votes

**2**answers

159 views

### Time estimate to determine if a number is prime [on hold]

How long does it take to verify that a given number is a prime number, as a function of its number of digits, in a personal computer, say? How computationally hard is this?

**11**

votes

**1**answer

463 views

### Is there a known primitive recursive upper bound on the nth “Zhang prime”

(This question is pure curiosity. Feel free to close it if you feel it is not appropriate for mathoverflow.)
In 2013 Zhang showed that there are infinitely many pairs of primes which are less that ...

**3**

votes

**1**answer

208 views

### “Weight-mondoromy” for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...

**2**

votes

**1**answer

132 views

### Special values of Hecke L-function

The Dedekind zeta function for a number field $K$ is defined as
$\zeta_K(s)=\sum_{I\subset O_K} (N_{K/\mathbb{Q}}(I))^{-s}$.
By attaching a Hecke character $\psi$, we can define ...

**3**

votes

**1**answer

120 views

### How to generate $n$ FP32 rationals s.t. no two distinct k-el. subsets have same sum?

First some
Background: I have lots and lots of integer matrices, whose rows are $k$-combinations (without repetitions and sorted) of numbers from the set $S:=\{1,...,n\}$ and needed to be compared ...

**3**

votes

**1**answer

218 views

### Waring's problem

What is the best known quantitative upper bound for the quantity $G(k)$?
I know that it's due to Trevor Wooley and in simplest form states that
$\limsup_{k \to \infty} \frac{G(k)}{k \log k} \le 1$.
...

**3**

votes

**1**answer

191 views

### Representation of GL(n, F_p) over F_p, for n small

The question is related to this post
Representation theory of the general linear group over a finite prime field
However, I am asking for more detailed references for n small, for example, for n=2, ...

**0**

votes

**0**answers

88 views

### Questions on roots of integral polynomials over $\mathbb{F}_p$

I asked earlier (A question on how polynomials split over $\mathbb{F}_p$ for large primes $p$) on when a polynomial $f(x) \in \mathbb{Z}[x]$ has a small root over $\mathbb{F}_p$, where $p$ is a large ...

**10**

votes

**4**answers

424 views

### How to calculate the infinite sum of this double series?

I'm calculating this double sum:
$$
\sum _{m=1}^{\infty } \sum _{k=0}^{\infty } \frac{(-1)^m}{(2 k+1)^2+m^2}
$$
I know the answer is
$$
\frac{ \pi \log (2)}{16}-\frac{\pi ^2}{16}
$$
which can be ...

**0**

votes

**0**answers

59 views

### Does this quaternary quartic form primitively represent infinitely many sufficiently large powers?

Let $g(x,y,z,t)=(x+y+z+t)^4-a h(x,y)$ where $h(x,y) \in \{x^4,xy^3,x^2y^2\}$
and $a$ is integer.
Does $g$ represent infinitely many powers $r^n$
with $n > 4$, $x+y+z+t,ah(x,y)$ take distinct ...

**23**

votes

**3**answers

718 views

### Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula.
I. Given the fundamental unit,
...

**7**

votes

**1**answer

533 views

### Roots of a polynomial in a finite field related to Fermat's Last Theorem

In my class, we proved the following condition: define the polynomial $P_l(x)$ as
$$P_l(x) = \sum_{r=1}^{l-1}{\frac{1}{r}x^{l-1-r}}$$
Then if for all $a \in \mathbb{Z}/l\mathbb{Z}-\{0,1\},$ ...

**0**

votes

**0**answers

67 views

### Almost locally stable properties of spaces [closed]

Assume that we are looking whether a Property $P$ holds for members $s$ from a space $X$.
Call a member $s$ of $X$, almost $\delta-$stable with respect to $P$ if property $P$ holds (or fails) for ...

**2**

votes

**2**answers

218 views

### Large solutions to Thue equations

Suppose that $f(x,y) \in \mathbb{Z}[x,y]$ is a homogeneous polynomial, or binary form, of degree $d$. The equation
$$f(x,y) = h$$
for a given integer $h$ is known as Thue's equation (so named ...

**23**

votes

**1**answer

891 views

### Which natural numbers are a square minus a sum of two squares?

Question: Which natural numbers are of the form $a^2 - b^2 - c^2$ with $a>b+c$?
This question came up in (Eike Hertel, Christian Richter, Tiling Convex Polygons with Congruent Equilateral ...

**1**

vote

**1**answer

221 views

### A question on how polynomials split over $\mathbb{F}_p$ for large primes $p$

Suppose that $f(x) \in \mathbb{Z}[x]$ is an irreducible polynomial (over $\mathbb{Q}$). Let $p$ be a very large prime with respect to the coefficients of $f$. Then it is possible that $f(x)$ may ...

**0**

votes

**0**answers

113 views

### Is the Jacobi theta function invertible?

Let $\theta$ denote the Jacobi theta function:
$$\theta=\sum_{k=0}^{\infty}{(-1)^kq^{k(k+1)}sin((2k+1)\frac{2\pi}{\omega_1}Re(z))},$$
and we have a complex number $t$. Suppose that we know there ...

**3**

votes

**0**answers

88 views

### Zeta functions with Brauer class

In algebraic geometry there are examples when a variety $X$ is somehow related (I call it double-mirror) to another variety $Y$, together with
a 2-torsion Brauer class. By "related" I mean statements ...

**6**

votes

**1**answer

232 views

### Euler's Triangular Number closure properties

Burton, in "Elementary Number Theory", states that the following problems are due to Euler 1775:
If $n$ is a triangular number, then so are $9n+1$, $25n+3$ and $49n + 6$.
R. F. Jordan in the J. ...

**3**

votes

**1**answer

172 views

+150

### Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form

(Note: This was cross-posted from MSE.)
Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$).
Therefore, ...

**4**

votes

**0**answers

125 views

### The divisors of $p-1$ and high-degree residues modulo $p$

Here is a somewhat more explicit version of a question that I asked a while ago.
Suppose that $p$ is a prime of the form $p=2n(n+1)+1$, with a positive integer $n$. Can every odd prime divisor of ...

**8**

votes

**1**answer

271 views

### minimizing an integral over integer-coefficient polynomials $\displaystyle \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $ [duplicate]

Let's consider the space $L^2[a,b]$ of functions on the interval and the norm:
$$ ||f(x)||^2 = \int_a^b |f(x)|^2 \, dx $$
Now what if we consider only polynomials with integer coefficients: $f(x) ...

**-1**

votes

**0**answers

68 views

### How do we usually calculate the size of the 2-division field of an elliptic curve over local fields

Let $E$ be an elliptic curve over $\mathbb Q$, with rank $0$ and $E(\mathbb Q)[2]=0$. Let $p$ be a prime number. How do we usually calculate $\# E(\mathbb Q_p)[2]$ when $E$ has good, bad reduction at ...

**0**

votes

**0**answers

66 views

### Lucasian Primality Criterion for Specific Class of $k \cdot 2^n-1$

Definition
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ , where $m$ and $x$ are nonnegative integers .
Conjecture
Let $N=k\cdot 2^n-1$ such ...

**2**

votes

**2**answers

120 views

### Number of *distinct* dot products of an integer vector by elements of a hyper-rectangle

Imagine a vector $\boldsymbol{v}$ composed of integers, and the set $S$ of all integer vectors within a hyper-rectange, with one corner at the origin and other at $\boldsymbol{m}$. In other words: $S ...

**6**

votes

**0**answers

153 views

### Can these two proofs of the parametrization of pythagorean triples be unified?

I am interested in the classical parametrization of rational solutions to $x^2 + y^2 = 1$.
One proof is the classical stereographic projection technique (see, e.g., here): Choose a rational point $P$ ...

**7**

votes

**2**answers

470 views

### Adjoining torsion points from abelian varieties

Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically ...

**7**

votes

**1**answer

271 views

### rational points of a hyperelliptic curve

I have the following hyperelliptic curve of genus $2$:
$$
y^2 = 561 x^6 - 41904 x^5 + 627264 x^4 + 11860992 x^3 - 197074944 x^2 + 124416^2
$$
I need to find all the rational points on this curve. ...

**1**

vote

**1**answer

188 views

### computing height on elliptic curve of the form $y^2=x^3-nx$

Let $E$ be the elliptic curve
$$y^2 =x^3 - 19*67 x$$
and $P=[26011/625,2159616/15625]$, I want to compute $\hat{h}(P)$ using formula given in
Fujita, Y., & Terai, N. (2011). Generators for the ...

**2**

votes

**1**answer

182 views

### Kloosterman sum zeroes

Can we prove the following statement about Kloosterman sums? Recall that a Kloosterman sum is given by:
$$K(a,b,m)=\sum_{0\leq x\leq m-1,\,\gcd(m,x)=1}e^{\frac{2\pi i}{m}(ax+bx^*)}$$
Where $x^*$ is ...

**10**

votes

**1**answer

1k views

### When does a Catalan number equal a Fibonacci number?

The $n=3$'rd Catalan number (A000108) is $1,1,2,5$ : $\frac{\binom{2n}{n}}{n+1}=\frac{\binom{6}{3}}{4}=\frac{20}{4}=$ 5.
The $n=4$'th Fibonacci number (A000045) is $1,1,2,3,5,...$ : 5.
Q. Which ...

**1**

vote

**0**answers

128 views

### Sets of coprime numbers

Consider the set $\{0, 3, 7, 15\}$ of four integers. If you add each of these numbers to a fixed power of 2, then the resulting four numbers are pairwise coprime. For example, $\{4, 7, 11, 19\}$ are ...

**31**

votes

**1**answer

1k views

### $\zeta(0)$ and the cotangent function

In preparing some practice problems for my complex analysis students, I stumbled across the following. It is not hard to show, using Liouville's theorem, that
$$\pi\cot(\pi ...

**1**

vote

**0**answers

45 views

### How can I interpolate between these sets of algebraic integers?

Consider the set $S_d(m)$ of algebraic integers whose minimal polynomials are of degree $\leq d$ and have constant and leading coefficients $+1$, and all other coefficients chosen from the set ...

**1**

vote

**1**answer

197 views

### How to test if the power of some algebraic number is the rational combination of two specific algebraic numbers?

Suppose we are given three algebraic numbers $\alpha,\beta,\gamma$ by presenting their minimal polynomial (degree less than $m$), the goal is to compute all positive integers $n$ such that $\alpha^n$ ...

**2**

votes

**1**answer

125 views

### Bounds on imaginary parts of partial Kloosterman sums?

For a prime $p$ and integers $a,m$, $0<a,m<p$ define the (partial Kloosterman) sum
$$ S_p(a,m) = \sum_{0<k<m} \exp\left(\frac{2\pi\mathrm{i}}{p}(a x + x^{-1})\right), $$
where $x^{-1}$ is ...

**4**

votes

**1**answer

169 views

### References to proofs of upper and lower bounds on the number of coprimes in an interval?

On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...

**3**

votes

**0**answers

194 views

### On the local root number(or local $\epsilon$-factor)

I want to ask some question related to the local root number.
Let $E/F$ be a quadratic extension of p-adic local fields and $\psi:E \to \mathbb{C}$ is an additive character of $E$.
Let $\phi:WD(E) ...

**4**

votes

**2**answers

417 views

### Twin primes for polynomials in $\Bbb Z[X]$

The following paper provides result on an analog of twin primes conjecture for $\Bbb F_q[X]$ http://www.math.uga.edu/~pollack/twins.pdf
Is there an analog of twin primes conjecture for $\Bbb Z[X]$?
...

**-4**

votes

**0**answers

37 views

### Find all solutions in positive integers of the diophantine equation $w^2+x^2+y^2=z^2$ [migrated]

It's an exercises of the text book Elementary Number Theory and It's Applications 6th Edition by Kenneth H.Rosen. I wanted to solve it using the method in solving the diophantine equation ...

**4**

votes

**0**answers

102 views

### Is this $S$-birational map an open immersion on its domain of definition?

My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...

**7**

votes

**2**answers

503 views

### Isomorphism problem for two radical extensions

Let $n\geq 2$ and let $a,b\in{\mathbb Q}$. Suppose that both the
polynomials $A=X^n-a$ and $B=X^n-b$ are irreducible. We want to know whether
( * ) there is a root $\alpha$ of $A$ and a root $\beta$ ...