More on $\phi$ in 2016

Number Theory Level 5

Find the minimum value of $k$ that makes $y$ an integer.

$\large{y= \log_{\phi(30^3)}\left(30^k \prod_{d \mid 30^3}\phi(d)\right)}$

Notation : $\phi(x)$ denotes the Euler totient function .

1 solution

Trevor Arashiro
Jan 9, 2016

If $x=\prod_{i=1}^{n} a_i^k$ where each $a_i$ is a distinct prime.

$\large{\displaystyle \prod_{d \mid x}\phi(d)=\left(\phi(x)\right)^{(k+1)^n-(k+1)^{n-1}}\times (x)^{\frac{k(k+1)^n}{2}-(k+1)^n+(k+1)^{n-1}}}$

Here $k=3,~n=3$

beat me to the solution(by a day)! you should prove it though(the claim).

Aareyan Manzoor - 5 years, 5 months ago