Find the number of solutions of the equation:
in the positive integers.
We set . It suffices to find the number of solutions of the equation
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
Comment: In general the equation
has solutions in the positive integers and solutions in the non negative integers.
We know that there are infinite Pythagorian triplets, that is numbers such that
Let us investigate if there exist triplets such that
where denotes the Euler’s totient function.
Indeed, there are infinite triplets such that is satisfied. For example noticing that
we deduce that for each natural such that we have
Let be positive numbers other than such that . Prove that
The Engels form of the Cauchy – Schwartz inequality gives us:
and the inequality is proven.
Evaluate the limit:
It is quite known that
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 .
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.