Problem 790 — Mathematical Challenge Section

Research Question

Find all real numbers $\alpha$ such that, for every $c > 0$, there exists an integer $n > 1$ satisfying

$$ \frac{\phi(n!)}{n^\alpha (n-1)!} > c. $$

Abstract

A number-theory problem published as Problem 790 in Pi Journal’s Mathematical Challenge section, concerning an expression involving factorials and Euler’s totient function.

What This Paper Does

The problem asks which real exponents make this expression unbounded as $n$ varies. Here $\phi(m)$ counts the positive integers at most $m$ that are coprime to $m$.

My Contribution

• Authored Problem 790 for the Mathematical Challenge section of Pi Journal.