## Parametric improper integral

Let such that . Prove that

## Improper Gamma integral

Let denote the Euler’s Gamma function. Prove that

where .

## Arithmotheoretic sum

Evaluate the limit:

**Solution**

However,

Hence

## Series of Bessel function

## Proof of “Fermat’s last theorem”

Let and . Prove that the equation

has no solution.

**Solution**

Without loss of generality , assume that . If held , then it would be thus . It follows from Bernoulli’s inequality that,

which is an obscurity. The result follows.