## Number of solutions

Find the number of solutions of the equation:

in the positive integers.

**Solution**

We set . It suffices to find the number of solutions of the equation

(1)

in the non negative numbers. We represent each sum of non negative integers with a sequence of dots () followed by a vertical bar (), after dots another one vertical bar etc, till we place the last dots ( without the vertical bar at the end.) For example the sum can be represented as

We note that every solution of matches a sequence that has dots in total and vertical bars. Conversely, every such sequence matches a solution of .

Thus, in total there are

solutions.

**Comment: **In general the equation

has solutions in the positive integers and solutions in the non negative integers.

## Eulerian equality

We know that there are infinite Pythagorian triplets, that is numbers such that

(1)

Let us investigate if there exist triplets such that

(2)

where denotes the Euler’s totient function.

**Solution**

Indeed, there are infinite triplets such that is satisfied. For example noticing that

we deduce that for each natural such that we have

## A logarithmic inequality

Let be positive numbers other than such that . Prove that

**Solution**

The Engels form of the Cauchy – Schwartz inequality gives us:

and the inequality is proven.

## A series limit

Evaluate the limit:

**Solution**

It is quite known that

Thus,

Therefore, by induction , for ,

Summing the last equation we get:

and raising the last equation to the power , we get:

Thus, by the squeeze theorem the limit is equal to .

## A limit on Dirichlet function

Let be the Dirichlet function;

Evaluate the limit

We simply note that

and the limit follows to be . The * reason *why

is because is rational if-f is a perfect square.