If $\phi(n)$ is the Euler-totient function, how can I show that $\phi(n) \ge \sqrt{n}$?
Asked
Active
Viewed 1,786 times
3
-
5$\varphi(2)=1<\sqrt2$. – Brian M. Scott Feb 20 '13 at 02:00
-
I'm sure this question has been asked, and answered, here, but I'm not sure how to find it. – Gerry Myerson Feb 20 '13 at 02:01
-
3$\varphi(6) = 2 < \sqrt{6}$ – davidlowryduda Feb 20 '13 at 02:01
-
1See, for example, http://math.stackexchange.com/questions/301837/is-the-euler-phi-function-bounded-below --- especially Brian's answer. – Gerry Myerson Feb 20 '13 at 02:08
-
@Gerry, http://math.stackexchange.com/questions/301837/is-the-euler-phi-function-bounded-below/301856#301856 – Will Jagy Feb 20 '13 at 02:08
-
It's true for $ n > \frac{81}{4} = 20.25,$ or $ n \geq 21.$ – Will Jagy Feb 20 '13 at 02:28
-
Hints: $\phi(n\cdot m)=\phi(n)\cdot\phi(m)$ if $n$ and $m$ are coprime, and $\phi(p^k) = p^{k-1}\cdot(p-1)$ for any prime number $p$ – Tobias Kienzler Oct 08 '13 at 11:04
1 Answers
9
Hint: both $\phi(n)$ and $\sqrt{n}$ are multiplicative, so it suffices to consider prime powers.
Robert Israel
- 448,999