**4**

votes

**1**answer

318 views

### A congruence conjecture regarding $(r-s)^4-1 \equiv 0\!\pmod{4r^2s}$

Is the following conjecture true?
Conjecture. If $r > s \ge 1$ are relatively prime integers such that
\begin{equation}
(r-s)^4-1 \equiv 0\!\pmod{4r^2s}, \tag{1}
\end{equation}
then $r-s = 1$ ...

**4**

votes

**1**answer

292 views

### searching for an elementary proof a complex analysis result

Given a function $ g $ entire on the whole complex plane $ C $, it is possible to find an entire function $f $ such that $ f(z+1) -f(z)=g(z) $. The proof can be given using riemann ...

**3**

votes

**1**answer

140 views

### Estimate sum with Euler function

(Note: this question was posted also in MSE)
I'd like to know if there's a closed formula or at least an estimate for the following (finite) sum:
$$
\sum_{D|p-1} \varphi(D) ...

**1**

vote

**0**answers

145 views

### Proofs for almost prime limits

A number $n$ with prime factorization $$n=\prod_{i=1}^rp_i^{a_i}$$
is a k-almost prime if it has a sum of exponents $$\sum_{i=1}^{r}a_i=k$$ i.e., when the prime factor (multiprimality) function ...

**3**

votes

**0**answers

89 views

### Lattices achieving best density

Let $\Lambda \subset \mathbb{R}^n$ be an Euclidean lattice with generator matrix $B$. Define the center density $\delta(\Lambda)$ in the usual way as $\delta(\Lambda) = \rho^n/|\det{B}|$, where $\rho$ ...

**9**

votes

**3**answers

2k views

### Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$

I'm working on solving the quartic Diophantine equation in the title. Calculations in maxima imply that the only integer solutions are
\begin{equation}
(r,s) \in \{(-3, -2), (-2, 3), (-1, 0), (0, ...

**3**

votes

**1**answer

267 views

### Is there an easy proof of this equation related to simultaneous Pell equations?

Working with the famous Baker-Davenport system of simultaneous Pell equations
\begin{align}
3x^2-2 &= y^2, &
8x^2-7 &= z^2, \qquad(\star)
\end{align}
I am left, after a series of ...

**11**

votes

**1**answer

515 views

### Are [Wieferich] primes the only solutions to the equation $2^{k-1} \equiv 1 \pmod{k^2}$?

While studying a certain Diophantine equation in the squarefree integer $k \ge 2$, I believe I have proven the necessary restriction
$$2^{k-1} \equiv 1\!\!\pmod{k^2}. \qquad(\star)$$
Based on what ...

**3**

votes

**0**answers

139 views

### elementary proof for existence of point with minimal period 2 for entire function

Fatou proved a very interesting result:
for a transcendental entire function $f$, the second itarate
$f^{2}$ has at least has one fixed point.
(Using the technique of Picard theorem)
This result ...

**6**

votes

**2**answers

605 views

### At what point would an elementary generalization of Bertrand's Postulate be interesting?

I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$.
At what point would an improvement on Nagura's result be interesting? ...

**25**

votes

**10**answers

3k views

### real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?

**8**

votes

**2**answers

714 views

### Length of Hirzebruch continued fractions

Suppose $a,b$ are two natural numbers relatively prime to $n$ and to each other. Assume $n\geq ab+1$. Suppose further that $\frac{a}{b}\equiv k \pmod{n}$ for some $k\in \lbrace 1,2,\dots, n-1\rbrace$ ...

**1**

vote

**1**answer

192 views

### On bounding the average cost of top-down merge sort

Let $A_n$ be the average number of comparisons to sort $n$ keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that
$$
A_0 = A_1 = 0;\quad A_n = ...

**31**

votes

**1**answer

1k views

### On a remark of Tait on FLT for the exponent 3

This is one of those recreational questions that aren't really about research. I found a curious remark in an old volume of American Mathematical Monthly (1922) which I'll quote below:
In the ...

**2**

votes

**3**answers

2k views

### Elementary proof of the equidistribution theorem

I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence $\alpha, 2\alpha, 3\alpha, \ldots \mod 1$ is ...

**12**

votes

**4**answers

2k views

### Is there an elementary way to find the integer solutions to $x^2-y^3=1$?

I gave this problem to my undergraduate assistant, as I saw that Euler had originally solved it (although I am having trouble finding his proof). After working on it for two weeks, we boiled the hard ...