First note that the only which fill the condition are

. We'll see there are no others.

Suppose fills the conditions, and let .

We can suppose now that and then arrive at a contradiction.

, so there exists a positive integer such that

Therefore and we can write

That is, .

Write . Plugging in the previous equality yields:

And since we have , so

Thus where we denote

We have and and these numbers

are coprime so . Furthermore,

and is coprime to , and ,

so , and thus

Therefore ,

contradicting our assumption that (see head of the post).