# Tagged Questions

**6**

votes

**2**answers

175 views

### Simple proof for $\sum_{i=1}^{n} a^{gcd(i,n)} $ is divisible by n

Burnside's Lemma Deduce That:
$$\sum_{i=1}^{n} a^{gcd(i,n)} $$
is divisible by n
it's a beautiful result. but i want to prove it without any abstract algebraic tools such as Burnside's Lemma...
is ...

**0**

votes

**0**answers

15 views

### Are all zeros of $\dfrac{\Gamma'}{\Gamma^2}(s) \pm \dfrac{\Gamma'}{\Gamma^2}(1-s)$ either real or on the line with $\Re(s)=\frac12$?

In this post it was proven that, except for a finite few, all zeros of $\Gamma(s) \pm \Gamma(1-s)$ are either real or reside on the line $\Re(s)=\frac12$.
I have been searching for similar reflexive $...

**5**

votes

**1**answer

305 views

### Uniqueness of sums of roots of unity

Let $\zeta:=e^{\frac{2\pi i}{n}}$, with $n\geq4$, and let $2\leq k\leq n-2$.
Let us suppose that the prime factorization of $n$ is $n=p_1^{\alpha_1}\cdot\dots\cdot p_s^{\alpha_s}$, with $\alpha_i>...

**2**

votes

**1**answer

276 views

### integral basis of orthogonal complement

Suppose there are $r$ linearly independent vectors $v_1,\dots,v_r\in \mathbb{R}^n$, all of them have integer-valued entries and $\|v_i\|_\infty\leq m$ for some integer $m$.
My goal is to find an ...

**23**

votes

**2**answers

1k views

### Monstrous Moonshine for Thompson group $Th$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,
$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...

**3**

votes

**1**answer

136 views

### Intuition behind the definition of $E_{\omega}$ in Wall's paper

Recently I am reading Wall's paper "On the Orthogonal Groups of Unimodular Quadratic Forms II". In this paper, I encountered with the map $E^1_\omega$, which now I am interested in.
Let $X$ be an ...

**0**

votes

**0**answers

75 views

### Summing a series of integrals [on hold]

EDIT: This IS related to my research (investigating representations of the harmonic mean)and I gave the wrong formula for the sum the first time around. The sum formula has been amended below.
I ...

**2**

votes

**1**answer

169 views

### Kummer extension of Galois modules

Let $k$ be a field of characteristic $p \geq 0$, $n$ an integer prime to $p$, and $x$ an element of $k \setminus \{0, 1\}$. I have read that the $n^{th}$ root of $1-x$ gives rise to a Galois module $E$...

**13**

votes

**2**answers

254 views

### Can something finite over $\mathbb{C}(q)$ be a modular form?

If $f\in\mathbf{C}[[q]]$ is non-constant, and algebraic over $\mathbf{C}[q]$ (in the sense that it is a root of a polynomial with coefficients in in $\mathbf{C}[q]$) then can $f$ be the $q$-expansion ...

**20**

votes

**1**answer

927 views

### What did Yu Jianchun discover about Carmichael numbers?

There's a news story going around (see for example [1]; other accounts are even more breathless) about an amateur mathematician, Yu Jianchun, finding an "alternative method to verify Carmichael ...

**2**

votes

**1**answer

84 views

### Examples of Sets with Positive Upper Density

While reading the statement of Roth's theorem I started asking myself what are examples of sets of positive upper density? It's not hard to come up with a few:
Flip a coin with probability $\mathbb{...

**17**

votes

**5**answers

3k views

### Proofs of Jacobi's four-square theorem

What are the nicest proofs of Jacobi’s four-square theorem you know? How much can they be streamlined? How are they related to each other?
I know of essentially three aproaches.
Modular forms, as ...

**16**

votes

**3**answers

1k views

### What proof of quadratic reciprocity is Hilbert referring to in this quote?

Let $(a, b)_v$ denote the Hilbert symbol on the completion $K_v$ of a global field $K$ at a place $v$. The Hilbert reciprocity law $\prod_v (a, b)_v = 1$ is a strict generalization of quadratic ...

**3**

votes

**0**answers

94 views

### A question about smooth convex lattice polygons

Let $P$ be a smooth convex lattice polygon in $\mathbb{R}^2$ (the lattice being $\mathbb{Z}^2$). Here smooth means that at any vertex of $P$, the two primitive integer vectors (i.e. vectors whose ...

**5**

votes

**3**answers

190 views

### 2-adic valuation of odd harmonic sums

(This question is cross-posted on math.stackexchange)
I'm playing with p-adic valuations, and find that the odd harmonic sums, $\tilde{H}_k=\sum_{i=1}^{k}\frac{1}{2i-1}$, has 2-adic valuation $||k^2||...

**3**

votes

**0**answers

228 views

### Arithmetical properties of certain recurrence relations

Consider the following recurrence relation:
$$(4i+6j+1)(2i+3j)a_{i,j}=3j a_{i+1,j-1}-2i a_{i-1,j}+32i(i-1) a_{i-2,j+1}, i\ge0, j\ge0,$$
$$a_{0,0}=1.$$
This equation appeared in the article (...

**3**

votes

**1**answer

126 views

### Asymptotics on the number of ways to pair off $\{1, 2, \dots, 2n\}$ into primes

Given $S = \{1, 2, \dots, 2n\}$, we can always pair off elements into $n$ pairs such that each sum to a prime. The proof of this fact is easy and follows from Bertrand's postulate.
Now, let $\gamma(n)...

**6**

votes

**3**answers

358 views

### linear independence of $\sin(k \pi / m)$

I have tried searching the literature for a result like the following, but have not found anything.
For a positive integer $m$, is it known that
$$\{ \sin (k \pi / m): 1 \leq k \leq m/2, (k,m)=1 \}$$
...

**5**

votes

**2**answers

244 views

### Is there any formula to find number of Pythagorean triplets between two integers 2 and j, j>2?

Given $j \geq 5$, is there a formula for the number of Pythagorean triplets $(a, b, c)$ satisfying the constraint that $a, b, c \leq j$?
There exists at least one Pythagorean triplet for $j\geq5$; ...

**0**

votes

**1**answer

153 views

### Characters and Galois stability

Let $G$ be a finite abelian group and $\widehat{G}$ the character group.
Let $S \subset \widehat{G}$ be a Galois-stable subset i.e. if $\chi \in S$, then the Galois conjugates $\chi^{\sigma} \in S$ ...

**3**

votes

**0**answers

158 views

### Artin conjecture on L-functions

Artin conjecture on Artin $L$-functions asserts that the Artin $L$-function $L(\rho,s)$ of a non-trivial irreducible representation $\rho$ of the Galois group $\Gamma$ of a number field admits ...

**6**

votes

**1**answer

686 views

### A general question about strictly non-palindromic numbers

For a definition, see the wikipedia page: http://en.wikipedia.org/wiki/Strictly_non-palindromic_number
So according to the wikipedia page, under properties, all strictly non-palindromic numbers with ...

**0**

votes

**0**answers

79 views

### What is the sharpest bound of this sum?

Fix $y\geq 1$ and let $\delta$ be a small enough positive real number. Put
$$\mathcal{D}^{+}=\left\{d=p_{1}...p_{l}: p_{l}<...<p_{1},\ p_{m} \leq y_{m} \ \textrm{for all odd} \ m \right\},$$
...

**9**

votes

**2**answers

493 views

### congruent number problem [closed]

I am studying the congruent number problem
and I heard that there is a paper by Kazuma Morita
which claims to solve this problem from my colleague.
I saw the paper on his homepage but it is very ...

**2**

votes

**1**answer

211 views

### How to count fixed-sized subsets of pairwise co-prime numbers less than a prime, satisfying an additional constraint?

In part of my research, I need to count (or find a polynomial bound for) the number of possible ways to select $n$ distinct integers less than the prime $p$, say $r_1, r_2, …, r_n$, which are ...

**-4**

votes

**0**answers

70 views

### How do I go about solving the following problem? [closed]

Given $(m_1,m_2, ...,m_r)\in Z^r_{\geq 0}$, and $a_1, a_2, · · · , a_r \in \mathbb{N}$ such that: $\sum_{i=1}^r a_im_i=qL$
where $L$ denotes the least common multiple of $a_1, a_2, · · · , a_r$ and $...

**2**

votes

**1**answer

104 views

### An upper bound for the number of prime numbers in non-linear progressions

Let $f(x)$ be a non-linear polynomial over $\mathbb{Z}$.
Consider the following sum $\pi_f(x):= \#\{y: y< x \text{ and } f(y) \text{ is prime} \}$.
Can we get an upper bound for $\pi_f(x)$?

**2**

votes

**0**answers

64 views

### Rank growth in ray class fields of primes that are inert in an imaginary quadratic extension

If $K=\mathbb{Q}[\sqrt{-d}]$, $E$ is an elliptic curve and $\operatorname{rank}(E(K))=\operatorname{rank}(E(\mathbb{Q}))$, are there infinitely many primes $l$ of $\mathbb{Q}$ so that:
(1.) $\...

**1**

vote

**0**answers

64 views

### Eigenvalues of the Laplacian modulo primes

Let $d\geq 1$ be an integer and consider $M = \mathbb{R}^d / \mathbb{Z}^d$. Let $\Delta$ be the Laplace operator on $M$. Then the eigenvalues are $$\frac{1}{4 \pi^2}\lambda_{\vec{k}} = |\vec{k}|^2.$$
...

**28**

votes

**2**answers

521 views

### $x_1 = 2$, $x_{n + 1} = {{x_n(x_n + 1)}\over2}$, what can we say about $x_n \text{ mod }2$?

This question was asked on MathStackexchange here, but there was no answer, so I am asking it here.
Let$$x_1 = 2, \quad x_{n + 1} = {{x_n(x_n + 1)}\over2}.$$What can we say about the behavior of $x_n ...

**1**

vote

**1**answer

256 views

### Some divisibility constraints in Frobenius coin problem

Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$.
Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ excellent if linear form ...

**-1**

votes

**1**answer

154 views

### When the Ratio of Two Factors is a power of $2$ (i.e. $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$)

We can write, $n!= 2^s \times a \times b$ where $gcd(a,b)=1$ and $2^{s+1} \nmid n!$.
Problem: Is there infinite number of $n$ when $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$?
Question:
1.How ...

**2**

votes

**3**answers

319 views

### Find a distinct postive integer solution to this $xyzw=504(x^2+y^2+z^2+w^2)$ diophantine equation

Following problem though not a research problem
if $x,y,z,w$ are postive integers,and such
$$xyzw=504(x^2+y^2+z^2+w^2)$$
such example $(x,y,z,w)=(21,63,84,84)$ hold,
Now My problem there exist ...

**2**

votes

**0**answers

118 views

### Spectral decomposition on GL(n)

If $\Delta_1, \ldots, \Delta_{n-1}$ are a basis of the ring of commuting bi-$SL(n,R)$-invariant differential operators, $L_0^2=L_0^2(SL(n,Z)\backslash SL(n,R))$ is the space of cuspidal automorphic ...

**9**

votes

**1**answer

400 views

### Constructing a family of convergents from continued fractions formed by a set of prime partial quotients

For a given real number $x$, the continued fraction representation $x = [a_0; a_1, a_2, \cdots]$ where $(a_n)_{n \geq 0}$is defined by setting $x = \alpha_0$, then $a_i = \lfloor \alpha_i \rfloor$, ...

**0**

votes

**0**answers

144 views

+50

### Deduction formula for Goldbach counting function

Assume $N\geq 1$ is integer and $P\geq 1$ is square-free integer. Goldbach counting function, $S_P(N,x)$, is defined to be the number of $n$ between 1 and $x$ such that $(N-n)(N+n)$ is co-prime to $P$....

**34**

votes

**1**answer

3k views

### Is the following identity true?

Calculation suggests the following identity:
$$
\lim_{n\to \infty}\sum_{k=1}^{n}\frac{(-1)^k}{k}\sum_{j=1}^k\frac{1}{2j-1}=\frac{1-\sqrt{5}}{2}.
$$
I have verified this identity for $n$ up to $5000$ ...

**0**

votes

**1**answer

169 views

### Prime quadratic non-residue

NC Ankeny showed assuming Riemann Hypothesis the least quadratic non residue( let it be '$r$') modulo some prime $p$ to be $O(\log^2 p)$. It is easy to see that $r$ is a prime.
I have following ...

**1**

vote

**0**answers

42 views

### How many rational points on $F(x,y)=m$ for homogeneous $F$?

Let $F \in \mathbb{Q}[x,y]$ be homogeneous of degree $d$ and $m$ is rational.
Assume the curve $C : F(x,y)=m$ is irreducible and of genus greater
than one.
Currently, how many rational points $C$ ...

**4**

votes

**1**answer

309 views

### Is the set $S:=\{n\mid n\in\mathbb{N}\text{ and $n$, $n+1$ and $n+2$ have the same number of divisors}\}$ infinite?

Is the set $S:=\{n\mid n\in\mathbb{N}\text{ and $n$, $n+1$ and $n+2$ have the same number of divisors}\}$ infinite?
Example: $33\in S$.

**5**

votes

**1**answer

190 views

### Numbers divisible only by primes of the form 4k+1

Let $A(N)$ denote the number of positive integers $n\le N$ composed of prime numbers $p\equiv 1\pmod 4$ only. Is there an asymptotic formula for $A(N)$ (as $N$ tends to infinity)?

**4**

votes

**1**answer

275 views

### Coefficients of factors of $x^n-1\in\mathbb{Q}[x]$

If you factor $x^n-1\in\mathbb{Q}[x]$, then for $n\leq 104$ the coefficients of the factors are in $\{-1, 0, 1\}$. (This is not true for $n=105$, however). Let $U$ be the set of positive integers $n$ ...

**7**

votes

**0**answers

119 views

### Efficient Dirichlet approximation (continued fractions?) over a number field

Is there an efficient algorithm for Dirichlet approximation for a given (high-degree) number field and its ring of integers, perhaps analogous to the Euclidean/continued fractions algorithm for the ...

**1**

vote

**0**answers

219 views

### $\mathfrak{q}$-ideal class bound

Let $K$ be a number field, $\mathcal{O}_K$ be its ring of integers. Let $\mathfrak{q}$ be a nonzero ideal in $\mathcal{O}_K$.
The $\mathfrak{q}$-ideal class group consists of equivalence classes of ...

**3**

votes

**1**answer

182 views

### Is there an infinite family of primes $q_{1},q_{2},…$ so that the rank of $E(\mathbb{Q}(\sqrt{-q_{i}}))$ equals that of $E(\mathbb{Q})$?

It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
Much less is known if $K$ is infinite-...

**-3**

votes

**0**answers

33 views

### problem related to conformal map of doubly connected region [closed]

Is the explicit result known?
Product[(1 - Cos[x]/Cosh[n*h]), {n, 1, Infinity}]

**0**

votes

**0**answers

66 views

### A question on indefinite binary quadratic forms

Let $f(x,y) = ax^2 + 2bxy + cy^2$ be an irreducible, indefinite (so that $b^2 - ac > 0$) binary quadratic form. Put $d = b^2 - ac$. We say that two pairs of integers $(x_1, y_1)$ and $(x_2, y_2)$ ...

**1**

vote

**0**answers

92 views

### Analog of Baker's theorem on linear combination of $\log a \log b$

Baker's theorem basically says that, given algebraic numbers $a_1,\ldots,a_n$ and $m_1,\ldots,m_n$, if there is no good reason for a linear combination
$$\sum m_i\log a_i$$
to cancel, then it is ...

**5**

votes

**0**answers

120 views

### Chowla's Construction of prime having least quadratic non-residue $\gg \log p$

This paper by NC Ankeny mentions that " S. Chowla has proved that there exist infinitely many primes $k$ where the
first $c_1\log k$ residues $(\bmod k)$ are all quadratic residues".
I recently ...

**1**

vote

**1**answer

183 views

### Complexity of $\mathsf{gcd}(a,b)\bmod N$

Given $a,b\in\Bbb N$ where each $a,b$ is $n$-bits, we can compute $\mathsf{gcd}(a,b)$ in $cn^{1+\epsilon}$ bit operations for some fixed $c\geq1$.
My query is given $N,a,b$ where $a,b$ is $n$-bits ...