**0**

votes

**0**answers

65 views

### Is this factoring algorithm sufficiently efficient for some integers of special kind?

Basically the question is if $m$ is factored over the integers,
can it be relatively efficiently factored over $\mathbb{Z}[\sqrt{n}]$
where $n$ is not factored, but might be of special form?
Suppose ...

**4**

votes

**0**answers

86 views

### Integer sum of distinct reciprocals with no integer subset sum

Question
$\def\nn{\mathbb{N}}$
For any $n \in \nn^+$, is there a finite set $S \subset \nn^+$ such that $\sum_{k \in S} \frac{1}{k} = n$ but $\sum_{k \in T} \frac{1}{k} \notin \nn^+$ for any $T ...

**6**

votes

**1**answer

237 views

### Definition of p-adic modular forms

I have been reading Hida's book "p-Adic automorphism forms on Shimura varieties" and I don't understand a point.
He first describes p-adic modular forms of tame level N as functions on the Igusa ...

**2**

votes

**1**answer

293 views

### Quadratic Diophantine equation in $\mathbb Z[T]$

I am trying to solve the following quadratic diophantine equation in $\mathbb Z[T]$:
$$((T+1)X+TY-1-Z)((T+1)X+TY-1+Z)=24XY$$
One has the following trivial solutions:
$(X,Y,Z)=(0,Y,\pm(1-TY))$, ...

**6**

votes

**1**answer

202 views

### A question about $(0,1]$-valued multiplicative functions

Suppose $f:\mathbb{N}\to [0,1]$ is a multiplicative function (i.e. $f(nm)=f(n)f(m)$ whenever $m$ and $n$ are coprime). Suppose $f$ has non-zero mean, which means
$$
\lim_{N\to\infty}\frac{1}{N} ...

**2**

votes

**2**answers

171 views

### Does the Divisor Function $\sigma(n)$ have analogues for other Fuchsian groups?

I have been reading about the divisor function $\sigma = 1 \ast 1$ and proved an elementary identity:
$$ \Big[\sum_{d|n} \sigma_0(d)\Big]^2 = \sum_{d|n} \sigma_0(d)^3$$
Here $\sigma_0 = \sum_{d|n} ...

**2**

votes

**1**answer

101 views

### Unique extension of the absolute value

Let $(K,u)$ be a complete valued field, $u$ be its discrete absolute value (corresponds to a discrete valuation on $K$), then:
($\ast)$ Let $E/K$ is a finite separable field extension, then the ...

**5**

votes

**1**answer

182 views

### Hyperelliptic curves with fixed genus and many rational points

It is a famous theorem of Faltings, previously a conjecture by Mordell, that any algebraic curve of genus at least $2$ defined over the rational numbers have at most finitely many rational points. A ...

**2**

votes

**0**answers

239 views

### A variant frobenius problem

From Sylvester's theorem we know that using only coins of sizes $a,b$, we can change exactly $\frac{(a-1)(b-1)}2$ different big coins up to $(a-1)(b-1)$.
Denote sets ...

**4**

votes

**2**answers

111 views

### Expected Cardinality of the First n Coefficients of a Continued Fraction

Is there anything known about the asymptotic expected growth of $card\{a_0,a_1,...,a_n\}$, where the $a_i$ are the first $n$ coefficient of a continued fraction $[a_0; a_1, ... ]$?

**4**

votes

**0**answers

134 views

### For which rings $A$ is $A-\{0\}$ Diophantine over $A$?

Let $A$ be a commutative ring. Recall that we call $S\subset A$ a Diophantine subset of $A$ if there exists a polynomial $P(t;x_1,\ldots,x_n)$ with coefficients in $A$ such that:
$$
t_0 \in S ~~ ...

**3**

votes

**1**answer

224 views

### On Heath-Brown's “Prime twins and Siegel zeros”

With a graduate student, I'm going through the paper (Proc. London Math. Soc. (3) 47 (1983), no. 2, 193–224.) We are quite baffled by the proof of Lemma 3 on p. 198.
Here's the background and ...

**6**

votes

**0**answers

166 views

### Criteria for irreducibility using the location of complex roots

I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...

**2**

votes

**1**answer

178 views

### Finite sequences which happen to be the sequence of orders of elements of a simple group

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation
$$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$
where $\phi$ is the Euler's totient function, $d$ ...

**-4**

votes

**1**answer

180 views

### Gauss-Wantzel theorem, Fermat primes and solvability of S_n [closed]

Gauss-Wantzel theorem asserts that a polygon with $n$ sides is constructible if and only if $n$ is a product of a power of $2$ and distinct prime Fermat numbers, where the Fermat number of index $k$ ...

**4**

votes

**0**answers

111 views

### Additivity of upper densities with respect to arithmetic progressions of integers

Let $\mathsf{d}^\star$ be the asymptotic upper density, defined on the power set of positive integers $\mathbf{N}^+$, so that
$$
\mathsf{d}^\star\colon \mathcal{P}(\mathbf{N}^+) \to\mathbf{R}\colon ...

**3**

votes

**1**answer

247 views

### Bateman-Horn conjecture, continued

In this question, we needed to compute the Hardy-Littlewood constant for the Bateman-Horn conjecture. Is there a simple argument to show that the infinite product actually converges? Also, is there a ...

**3**

votes

**2**answers

260 views

### Is it normal surface of general type to have infinitely many positive rank elliptic curves?

Cross-posted from MSE.
I am not good at algebraic geometry and almost surely am
misunderstanding something.
Got an alleged argument against Bombieri-Lang conjecture and
would like to know what the ...

**1**

vote

**0**answers

83 views

### Question about expression of a sum of two Hecke eigenvalues

I did some computations but I am stuck in finding the exression of the sum
$$\lambda_f(n^2)+\lambda_f(n)^2 $$ in terms of $\lambda_f(n),$ where $f$ is a modular form for the full modular group. Any ...

**5**

votes

**2**answers

320 views

### Finding the inertia group

Set $h(x) = x^5+x^4+x^3+x^2+x-1$, let $L$ be the splitting field of $h$ over $\mathbb{Q}$, and let $p$ be a prime of $L$ lying over $2$.
What is the isomorphism class of the inertia group $I_p$, ...

**3**

votes

**0**answers

153 views

### The sum of greatest common divisors

Let
$$
G_n=\sum_{i=1}^n\sum_{j=1}^n \gcd(i,j).
$$
What is the asymptotics of $G_n$? (As much of it as it's known to date, preferably.)

**6**

votes

**0**answers

173 views

### The p-adic valuation of a linear recurrence

Let $(u_n)_{n \geq 0}$ be an integer-valued linear recurrence of order $k \geq 1$. Precisely,
$$u_n = a_1 u_{n-1} + \cdots + a_k u_{n - k} \quad \forall n \geq k ,$$
for some $a_1, \ldots, a_k \in ...

**2**

votes

**4**answers

276 views

### Prime divisors of values of a polynomial on an infinite set

This may be a well known problem:
Let $f$ be a polynomial with integer coefficients. Is it possible for the set of prime divisors of values of $f$ on an infinite set of integers, to be finite?
I ...

**1**

vote

**1**answer

182 views

### Will a tilting sheaf over Z which is a generator over Q be a generator modulo every prime?

Assume you have a smooth quasi-projective scheme $X$ (you can actually assume $X$ is projective over an affine scheme of finite type) defined over $\mathbb Z$ (or if you prefer, a discrete valuation ...

**4**

votes

**0**answers

235 views

### Motivic fundamental group of the moduli space of curves?

Suppose I have a smooth projective family of varieties of varieties over $\mathcal M_g$ - i.e. a universal functor, commuting with deformations, from curves to smooth projective varieties. Can I ...

**17**

votes

**0**answers

394 views

### Which sets of roots of unity give a polynomial with nonnegative coefficients?

The question in brief: When does a subset $S$ of the complex $n$th roots of unity have the property that
$$\prod_{\alpha\, \in \,S} (z-\alpha)$$
gives a polynomial in $\mathbb R[z]$ with ...

**2**

votes

**0**answers

130 views

### Table of LCM's vs. table of products

In 2004 Kevin Ford established sharp asymptotics on Erdős' problem on the number of different products $a\cdot b$, $a,b\in \{1,\dots,n\}$.
(http://arxiv.org/abs/math/0401223, see also discussion ...

**0**

votes

**1**answer

95 views

### Discussion for the sign of a specific sum

Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$
I want to ...

**4**

votes

**1**answer

363 views

### Unexpectedly prime rich cubic polynomial

We got a cubic polynomial which is unexpectedly prime rich.
Let $f(x)=29160 x^3 + 30132 x^2 + 8046 x + 643$ and
$\pi_f(n)$ the number of primes values of $f(x)$ for $x \in [1,n]$.
Let ...

**1**

vote

**1**answer

82 views

### Values of the Kronecker symbol of recursvie sequences

Is there something known about the values of the Kronecker symbol $\left(\frac{a_n}{a_{n+1}}\right)$ if $a_n$ is a recursive sequence? I know something about this if $a_n$ is the Fibonacci sequence or ...

**0**

votes

**1**answer

66 views

### Expression of a sum of Hecke eigenvalues in terms of one Hecke eigenvalue

Let $f$ be a modular form of an even weight $k$ over the modular group $SL_2(Z).$ Denote $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ I am doing some calculations and I am stack in ...

**4**

votes

**0**answers

137 views

### Expressing every algebraic number using roots of trinomials?

This question is a continuation of Is every polynomial a factor of a trinomial?
We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m ...

**2**

votes

**0**answers

142 views

### Help for reference of moduli stack of fake elliptic curve

I see everywhere says the following:
Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$. ...

**8**

votes

**2**answers

364 views

### Inverse Galois problem for $GL_2$ of a compact local ring

Let $A$ be complete noetherian local ring with maximal ideal $m$ and residue field $A/m$ a finite field (in other words, $A$ is a noetherian compact local ring)
For which $A$ as above is there a ...

**5**

votes

**1**answer

377 views

### Is every polynomial a factor of a trinomial?

We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m + C$ for some $n \geq m \in \mathbb{N}$.
Is it true that for each ...

**6**

votes

**2**answers

329 views

### How simple does a $\mathbb{Q}$-simple group remain after base change to $\mathbb{Q}_{\ell}$?

Of course the general answer to the question in the title is: not very simple.
I could not think of a better title, so let me explain my question in more detail.
I have a number field $E/\mathbb{Q}$, ...

**0**

votes

**0**answers

66 views

### Polynomial identities for congruent numbers and Bunyakovsky's conjecture

Bunyakovsky's conjecture states that polynomial with integer coefficients
takes infinitely many prime values unless there are obvious reasons not
to.
It appears to imply something about polynomial ...

**10**

votes

**0**answers

338 views

### Between Fermat's primes and the twin primes

Let me start with a curiosity. The integers $11,13,17,19$ are prime numbers, and $101,103,107,109$ are prime as well. One might wonder whether there is another occurrence where $10^m+1,10^m+3,10^m+7$ ...

**4**

votes

**0**answers

174 views

### On $a+b+c= abc = n$, elliptic curves, and solvable Galois groups

Solving $a+b+c = abc = 6$ in the rationals entails solving,
$$-24a+36a^2-12a^3+a^4=z^2\tag1$$
which is birationally equivalent to an elliptic curve. It can be shown that if $a$ is a solution, then ...

**0**

votes

**0**answers

105 views

### Representation as sum of squares

Given $N\in\Bbb N$ such that $\prod_{i=1}^mp_i=N$ with $p_i$ being similar sized primes such that $p_i\neq p_j$ if $i\neq j$ where $m\in[1,\log\log N]$, consider ...

**0**

votes

**0**answers

36 views

### Relation between $\Gamma$-percuspidal parabolic subgroups and split parabolic subgroups of real semisimple Lie groups

Let $G$ be a reductive algebraic group defined over $\mathbb{Q}$. Let $\Gamma$ be a lattice in $\mathcal{G}:= G(\mathbb{R})$. I am interested in knowing under what conditions on either of ...

**1**

vote

**0**answers

97 views

### In how many ways can one extend the zero section of the affine line with a double origin

Let $X$ be the affine line with a double origin over Spec $\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over Spec $\mathbb Q$.
Let $0$ be one of the origins of ...

**5**

votes

**0**answers

106 views

### Minimal possible cardinality of a $(a_1, …, a_k)$-distributable multiset

Suppose we have a multiset $M$ of positive rational numbers. Sum of $M$ equals $1$. We'll call this multiset $n$-distributable for some $n\in \mathbb{N}$, if there exists a partition $M_1 \sqcup ... ...

**1**

vote

**1**answer

153 views

### Central binomials and irrationals

I suppose this is une cause perdue, but it would be nice if the following held.
Let $\theta$ be an irrational, and let $c_m = {{2m}\choose m}$, the central binomial. For a real number, and let $d(r)$ ...

**12**

votes

**2**answers

490 views

### Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes

Is there an explicit infinite set of primes, modulo which $X^5 - X - 1$ is irreducible?
Since our polynomial's Galois group over $\mathbb{Q}$ is $S_5$, Chebotarev's density theorem implies that ...

**3**

votes

**2**answers

448 views

### Fermat's Last Theorem “$\pm k$”

Let $\mathbb{N}=\{1,2,3,\ldots\}$ be the set of positive integers. For $n,k\in\mathbb{N}$ we define $$\text{Sol}(n,k) = \{(a,b,c)\in \mathbb{N}^3: |a^n + b^n - c^n| \leq k\}.$$ (The set ...

**0**

votes

**0**answers

81 views

### Explicit formula for the product of three Hecke eigenvalues

I am interested in Hecke eigenvalues $\lambda_f(n)$ which are normalized Fourier coefficients of modular forms of an even weight $k$ for the full modular group. I often know that these coefficients ...

**1**

vote

**1**answer

422 views

### Partitioning $2p$, subject to a divisibility condition.

Let $p$ be a prime number and $l$ be the greatest prime number less than $2p$. Moreover, let $m,t_i<p$ be positive integer numbers such that $\sum_{i=1}^mit_i=2p$. Is it possible that ...

**0**

votes

**0**answers

62 views

### Exists $n$ where multisets of nonzero digits of $n$, $2n$, …, $2000n$ are equal?

As the question title suggests, does there exist a positive integer $n$ such that the multisets of nonzero digits of each of the numbers $n$, $2n$, ..., $2000n$ are equal?

**9**

votes

**3**answers

458 views

### Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
x^{-2j} & -x^{2j+1} \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
...