For any prime p and any positive integer n, let \(\nu _p(n)\) be the largest integer \(\alpha \) such that \(p^\alpha \mid n\) . Let \(\varphi \) be Euler’s totient function. In this paper, we prove that \(\nu _p(\varphi (n))\le \lfloor \log _pn\rfloor \) for all primes p and all positive integers n, where \(\lfloor x\rfloor \) denotes the largest integer not greater than x, and give the sufficient and necessary conditions for equality. In particular, \(\nu _2(\varphi (n))=\lfloor \log _2n\rfloor \) if and only if n is the product of distinct Fermat primes. Moreover, we also prove that there are infinitely many pairs (p, n) such that \(\nu _p(\varphi (n))=\lfloor \log _pn\rfloor \ge 1\) , and for a fixed p, the set of positive integers n with \(\nu _p(\varphi (n))=\lfloor \log _pn\rfloor \) has asymptotic density zero. Two conjectures are posed for further research.
