# Tagged Questions

**0**

votes

**0**answers

19 views

### Groupes fondamentaux de Tate mixte

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 ...

**4**

votes

**2**answers

742 views

### Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem:
Exponentiating Polynomial Root Problem (EPRP)
Let $p(x)$ be a polynomial with $\deg(p) \geq ...

**5**

votes

**3**answers

161 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 ...

**26**

votes

**7**answers

8k views

### Collecting proofs that finite multiplicative subgroups of fields are cyclic.

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...

**3**

votes

**0**answers

114 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

**1**answer

92 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 ...

**11**

votes

**1**answer

611 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

**1**answer

137 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 ...

**24**

votes

**3**answers

2k views

### Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...

**8**

votes

**1**answer

524 views

### Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...

**44**

votes

**4**answers

1k 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 ...

**5**

votes

**1**answer

265 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} & > & ...

**5**

votes

**3**answers

658 views

### moduli interpretations for modular curves

Some big picture questions -
What are some applications of the moduli interpretation for congruence curves? Specifically, the interpretations for congruence curves parametrizing elliptic curves with ...

**28**

votes

**2**answers

3k views

### Is Li(x) the best possible approximation to the prime-counting function?

The Prime Number Theorem says that $\lim_{n \to \infty} \frac{\pi(n)}{\mathrm{Li}(n)} = 1$, where $\mathrm{Li}(x)$ is the Logarithm integral function $\mathrm{Li}(x) = \int_2^x \frac{1}{\log(x)}dx$. ...

**2**

votes

**2**answers

318 views

### Primes as uncorrelated random variables [on hold]

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 ...

**0**

votes

**1**answer

273 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**

votes

**1**answer

413 views

### Number of solutions in a sum of squares Diophantine equation

Let $n$ be an integer. I'm interested in upper bounding the number of integer solutions of
\begin{equation*}
x_1^2+x_2^2+\ldots+x_p^2=y_1^2+y_2^2+\ldots+y_p^2,
\end{equation*}
where for all ...

**4**

votes

**1**answer

301 views

### Counting number of points in a lattice with bounded sup norm

Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let
$\| \bar{x} \| = max_{1 \leq i \leq n} \{ |x_i| \}$, i.e. the sup norm. Let $\lambda_1, ..., \lambda_n$
be a ...

**12**

votes

**1**answer

1k views

### A question about Iwasawa Theory

I am just reading about Iwasawa theory about Coates and Sujatha's book on Iwasawa Theory. I was wondering that since Iwasawa thought about the whole theory from the analogy of curves over finite ...

**7**

votes

**1**answer

265 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 ...

**3**

votes

**4**answers

302 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 ...

**-2**

votes

**0**answers

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 ...

**7**

votes

**0**answers

148 views

### When is the Locus of Equi-modular points of two monic polynomials with integer coefficients contained in the unit disk?

If $\lambda_{1}(z)$ and $\lambda_{2}(z)$ are two monic polynomials (relatively prime) with integer coefficients and $$\Gamma=\lbrace z \rm{\ s.t.\ } |\lambda_{1}(z)|=|\lambda_{2}(z)|\rbrace,$$ when is ...

**3**

votes

**0**answers

47 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 ...

**24**

votes

**3**answers

474 views

### Ordering subsets of the cyclic group to give distinct partial sums

Suppose that you are given a set $S$ of $k$ nonzero elements from $\mathbb{Z}_n$. Is it always possible to order the elements of $S$, say $a_1,a_2,\dots,a_k$ in such a way that the partial sums ...

**5**

votes

**1**answer

162 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 $\ell$.
...

**-1**

votes

**0**answers

69 views

### Are Modular Collatz Graphs strongly connected? [on hold]

A while ago, I stumbled on the idea of representing the Collatz function, modulo a prime $p$, as a directed graph. Define, as usual
$$
T(x) = \begin{cases}
(3x+1)/2 & \text{if $x$ is odd,} \\
x/2 ...

**4**

votes

**1**answer

138 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 ...

**14**

votes

**6**answers

2k views

### Proofs of the Chevalley-Warning Theorem

A well known proof of the Chevally-Warning Theorem is the one listed on wikipedia: http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem
Are there any other proofs of this, or ...

**13**

votes

**1**answer

414 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**

votes

**3**answers

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 ...

**7**

votes

**1**answer

365 views

### What does the sum of the reciprocals of all the highly composite numbers converge to?

I've calculated the sum of the reciprocals of all the $156$ first highly composite numbers up to $10^{18}$:
$\sum \dfrac{1}{HCC(n)} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{1}{12} + ...

**1**

vote

**0**answers

245 views

+100

### 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 ...

**11**

votes

**2**answers

814 views

### Which recursively-defined predicates can be expressed in Presburger Arithmetic?

In Presburger Arithmetic there is no predicate that can express divisibility, else Presburger Arithmetic would be as expressive as Peano Arithmetic. Divisibility can be defined recursively, for ...

**2**

votes

**2**answers

566 views

### inverted factorial and trailing zeros problem

First than anything a big Hello for all math fans like me.
I've found a problem that is pretty interesting and I can't find the answer.
As all of you must know, to counting the trailing zeros of $n$ ...

**22**

votes

**17**answers

5k views

### Applications of finite continued fractions

I know some applications of finite continued fractions. Probably you know more. Can you add anything? (For for Applications of periodic continued fractions I have made a special topic.)
1) (Trivial) ...

**4**

votes

**1**answer

132 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 ...

**11**

votes

**2**answers

446 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 ...

**4**

votes

**2**answers

587 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, ...

**-2**

votes

**0**answers

41 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**

votes

**2**answers

293 views

### Brocard's problem [closed]

According to Brocard's problem
$$x^{2}-1=n!=4!*(5+1)(5+2)...(5+s)$$
here,$(5+1)(5+2)...(5+s)=\mathcal{O}(5^{r}),4!=k$. So,
$$x^{2}-1=k *\mathcal{O}(5^{r})$$
Here, $\mathcal{O}$ is Big O ...

**15**

votes

**1**answer

519 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

**1**answer

205 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 ...

**10**

votes

**0**answers

176 views

### Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...

**6**

votes

**0**answers

94 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):
...

**14**

votes

**1**answer

996 views

### The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13

When $N > 1$, the modular forms $\Delta(z)$ and $\Delta(Nz)$ are algebraically independent over the complexes, and the same then is true of their expansions at infinity. But using the fact that
the ...

**26**

votes

**5**answers

2k views

### The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$

I was asked the following question by a colleague and was embarrassed not to know the answer.
Let $f(x), g(x) \in \mathbb{Z}[x]$ with no root in common. Let $I = (f(x),g(x))\cap \mathbb{Z}$, that is, ...

**6**

votes

**2**answers

547 views

### Counterexample to Proposition of Granville related to abc conjecture

Looks like there is counterexample to Proposition related to
abc conjecture. Confusion is likely.
From RATIONAL AND INTEGRAL POINTS ON QUADRATIC TWISTS OF A GIVEN HYPERELLIPTIC CURVE, Andrew ...

**4**

votes

**0**answers

130 views

### Simultaneously using the real and 2adic norms

In the book Modern Computer Arithmetic, there is a section that talks about division with remainder and such in a way that exploits the interplay between the real and 2-adic norms; e.g. the linked-to ...

**16**

votes

**1**answer

899 views

### Has this strong number theoretic conjecture of Euler been proved, and where could I find such a proof?

Polya cites this work of Euler as an example of a conjecture which Euler considered impossible to doubt, and yet still needing a demonstration.
It is on pages 90-98 of "Induction and Analogy in ...