# Tagged Questions

**3**

votes

**0**answers

118 views

### P-adic Weierstrass Lemma for several variables

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...

**12**

votes

**1**answer

308 views

### Permanent of a matrix of odd integers

It is clear that the permanent of an $n\times n$ matrix which entries are odd integers, is an even number, as it is the sum of $n!$ odd numbers. I am interested in finding the highest power of $2$ ...

**6**

votes

**1**answer

222 views

### Algebraic integers in skew fields

Hi everyone,
let $D$ be a skew field, which is finite dimensional over its center $k$. Assume that $k$ is a number field, and let $\mathcal{O}_D$ be the set of elements $z\in D$ which are roots of a ...

**1**

vote

**2**answers

484 views

### How to interpret this class of numbers?

Let's say $ f(p) $ is a number defined as shown below:
$ \hspace {10 mm}f(p) = {\sqrt {2} ^ {{\sqrt {2} ^ \sqrt {2}} ^ {... \hspace {1 mm} p\hspace {1 mm} times } }} $
What I understand is:
We ...

**6**

votes

**0**answers

511 views

### Hard divisibility problem [closed]

The statement of this problem is elementary... How about the proof ?
Let $p$ be an odd prime number. For every integer $a$ , define the number
$ ...

**11**

votes

**2**answers

649 views

### Interesting result on the Euler-Maschroni constant - what is the background?

Today I entered the following expression in maple:
$$a_i = H_{10^i} - ln(10^i) - \gamma$$
Here $H_j$ equals $\sum_{k=1}^{j} 1/k$ and $\gamma$ is the Euler-Mascheroni constant.
When I computed $a_n$ ...

**0**

votes

**1**answer

421 views

### Maximum sum of 3 consecutive numbers in a permutation [closed]

Given that $X = \{0, 1, 2, ..., 7, 8, 9\}$, and $P$ is a permutation on $X$. Let $M(P)$ be the maximum sum of 3 consecutive elements. For example, if $P = (0, 2, 4, 1, 5, 7, 9, 3, 8, 6)$, then $M(P)$ ...

**11**

votes

**4**answers

820 views

### Applications of full integral weight modular forms in elementary number theory

Except for Eisenstein series having the divisor functions as their Fourier coefficients, is there any other full integral weight modular form (of some level, preferably full) having arithmetic ...

**5**

votes

**4**answers

498 views

### Logical equivalences for FTA

I hope this isn't a stupid question...
It's well known that (in the presence of various other axioms), Euclid's Postulate 5 ('parallel axiom') is equivalent to the Pythagorean Theorem. That is, ...

**33**

votes

**6**answers

2k views

### What is the simplest, most elementary proof that a particular number is transcendental?

I teach, among many other things, a class of wonderful and inquisitive 7th graders. We've recently been studying and discussing various number systems (N, Z, Q, R, C, algebraic numbers, and even ...

**1**

vote

**0**answers

332 views

### Statistic of cubic Irreducible polynomial with cyclic Galois group

Van der Waerden proved for monic irreducible polynomials $f$ of degree $n$ with bounded height $X$:
$$
|\{ f(x)\in \mathbb{Z}[x]: Gal(f)\neq S_n \}|=o(X)
$$
Where height of $f$ is defined by maximum ...

**2**

votes

**0**answers

459 views

### integer solutions of (n!+1)=m^2

Consider $4!=24$, if you add one you get $25=5^2$. The same occurs with $5! = 120 = 11^2 - 1$, and $7! = 5040 = 71^2 - 1$. Are there other solutions of the equation $n!+1 = m^2$?
I verified that no ...

**20**

votes

**2**answers

1k views

### What is the L-function version of quadratic reciprocity?

Quadratic reciprocity theorems states that for two different odd prime p and q,
we have (p/q)(q/p)=(-1)^(p-1)(q-1)/4.
What is the statement of this theorem in L-function?

**0**

votes

**1**answer

254 views

### Estimating the probability of co-occurrence of a set of positive integers

How to estimate the probability of co-occurrence of the positive integers $c_i$ and $d_i$, $1 \leq i \leq t$ drawn from the uniform range $1$ to $2^k-1$, such that $\Sigma^t_{i=1} c^2_i = ...

**4**

votes

**2**answers

691 views

### When does a a rational function have infinitely many integer values for integer inputs?

Consider rational functions $F(x)=P(x)/Q(x)$ with $P(x),Q(x) \in \mathbb{Z}[x]$. I'd like to know when I can expect $F(k) \in \mathbb{Z}$ for infinitely many positive integers $k$. Of course this ...

**2**

votes

**0**answers

611 views

### Strong Bezout's Identity?

Let $\{ a_i \}_{i=1}^N $ be a set of elements of the ring of integers, $\mathbb{Z}_D$ and define $g = \text{gcd}(a_1, a_2,\ldots, a_N, D)$. Then Bezout's Identity states that there exists another set ...

**4**

votes

**1**answer

2k views

### Sum Equals Product

Sum Equal Product
There are many articles in re this issue on the Web, although many are restricted to special cases (e.g., John Cook’s sum of tangents = product of tangents), and even some involving ...

**0**

votes

**2**answers

958 views

### Arctangents and the golden ratio

Why is the golden ratio lurking in $(d/dx)\arctan\left( x + \frac{1}{x} \right)$
$$
= \frac{\left(\frac{1+\sqrt{5}}{2}\right)}{x^2 + \left(\frac{1+\sqrt{5}}{2}\right)^2} + ...

**21**

votes

**3**answers

1k views

### Proving non-existence of solutions to $3^n-2^m=t$ without using congruences

I made a passing comment under Max Alekseyev's cute answer to this question and Pete Clark suggested I raise it explicitly as a different question. I cannot give any motivation for it however---it was ...

**8**

votes

**2**answers

539 views

### cocompact discrete subgroups of SL_2

How can one construct families of cocompact discrete subgroups of the topological group $\text{SL}_2(\mathbb{C})$?
Here quaternion algebra's might help, I believe, but I have some difficulties with ...