Sylvester and Schur Theorem says that for $x > n$, there is always at least one integer with a prime factor $> n$.
Bertrand's Postulate shows there is at least one and provides a method for determining a minimum number that increases as $n$ increases.
Bertrand's Postulate's proof depends on calculus to show that a function is increasing.
If my logic is sound, I have found a very elementary argument that relies on nothing more advanced than the Arithmetic Mean/Geometric Mean and argues that this is true for $n\ge 11$ (if I did my analysis correctly).
If this is already well known, could someone post a citation? If this is easily shown to be incorrect, could you provide the reasoning or counter example.
I was able to find this very interesting paper by Shorey and Tijdeman. (It takes me a long time to digest these papers so I apologize if the answer to my question is obvious from reading this paper.)
Is there any other such paper that would be interesting? I am trying to see how my method compares to what is well known.