3
votes
0answers
116 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
1answer
297 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
1answer
218 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
2answers
475 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
0answers
504 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
2answers
644 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
1answer
406 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
4answers
801 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
4answers
494 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, ...
32
votes
6answers
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
0answers
328 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
0answers
416 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 ...
18
votes
2answers
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
1answer
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
2answers
651 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
0answers
602 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 ...
3
votes
1answer
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
2answers
942 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} + ...
7
votes
3answers
2k views

How to find/guess a polynomial sequence?

My question is motivated by the recent question and more recent appearance of its author Bruce Westbury. Most of you know that the best way to find a sequence of integers is looking for it on The ...
20
votes
3answers
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
2answers
523 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 ...