Irrational number

Prove that the number $\newcommand{\lcm}{{\rm lcm}}$

$$\mathcal{I}=\sum_{n=1}^{\infty} \frac{1}{\lcm (1, 2, \dots, n)}$$

is irrational.

Solution

The sum is $2$

Let $n$ be a perfect number. Prove that:

$$\mathbf{\sum_{d \mid n} \frac{1}{d}= 2} $$

Solution

Inverse sum of amicable numbers

Let $m , \; n$ be a pair of amicable numbers. Prove that:

$$\mathbf{\left ( \sum_{d \mid m}\frac{1}{d} \right )^{-1}+ \left ( \sum_{d \mid n} \frac{1}{d} \right )^{-1}=1}$$

Solution

$\varphi(n) \mid n$

Find all $n \in \mathbb{N}$ such that $\varphi(n) \mid n$.

Solution

It is an integer

Prove that the expression

$$\frac{\gcd(m,n)}{n} \binom{n}{m}$$

is an integer, where $m, n \in \mathbb{N}$.

Solution

The number is odd

Let $\phi$ denote Euler's $\phi$. If $\phi(n)=\phi(2n)$ then prove that $n$ is odd.

Solution

A determinant

Let $(a, b)$ denote the great common divisor of the numbers $a, b$. Prove that:

$$\begin{vmatrix}
\left ( 1,1 \right ) &\left ( 1,2 \right )  &\cdots  &\left ( 1,n \right ) \\
 \left ( 2,1 \right )&\left ( 2,2 \right )  & \cdots &\left ( 2,n \right ) \\
 \vdots & \vdots  & \ddots & \vdots \\
 \left ( n,1 \right )& \left ( n, 2 \right ) &\cdots  & \left ( n,n \right )
\end{vmatrix} = \prod_{k=1}^{n}\phi(k)$$

where $\phi$ is Euler's phi function.

Solution

On Mobius function

Let $n \geq 3$. Prove that:

$$\sum_{k=1}^{n} \mu(k!)=1$$

Solution

Congruency of binomial coefficient

Let $p$ be a prime number such that $n<p<2n$. Prove that:

$$\binom{2n}{n} \equiv 0 (\bmod p)$$

Solution

Transcedental roots

Prove that the non zero roots of the equation $\tan x =x$ are transcedental.

Solution

Irrational number

Let $N>1$. Prove that the number:
$$\mathcal{N}= \sqrt{1\cdot 2 \cdot 3 \cdot 4 \cdots (N-1)\cdot N}$$

is irrational.

Solution

Limit and number theory

Let $\varphi $ denote the Euler function. Evaluate the limit:

$$\lim_{n \rightarrow +\infty} \frac{1}{n^2} \sum_{k \leq n} \varphi(k)$$

Solution

Let us copy ... Fourier

Let $a_n$ be a strictly increasing sequence of positive integers. Prove that

$$x_n =\frac{1}{a_1} + \frac{1}{a_1 a_2} +\cdots + \frac{1}{a_1 a_2 a_3 \cdots a_n}$$

converges to an irrational number.

Solution

Diophantine equation and triplets

Find five (5) triplets $(x, y, z)\in \mathbb{N}^3$ such that $\gcd(x, y, z)=1$ and
$$x^3+y^3 = 3xyz$$

Solution

Transcedental number

Examine whether or not the number $\displaystyle \sum_{n=1}^{\infty} \frac{1}{2^{n^2}}$ is transcedental.

Solution