More problems in 2016 part 8

If $\large{x=\displaystyle \prod_{i=1}^{n} (a_i)^{k_i}}$ where each $a_i$ is a distinct prime number, then $\phi(x)=x\displaystyle \prod_{i=1}^{n} \left(1-\frac{1}{a_i}\right)$

This function is also known as the totient function.

Since 1729 can be factored into 3 distinct prime factors, we have

$\displaystyle \prod_{d \mid 1729}\frac{1}{\phi(d)}=\frac{1}{\phi(1729)^m}$

Where $m=\frac{1}{3}\binom{3}{1}+\frac{2}{3}\binom{3}{2}+1\binom{3}{3}=4$

Thus $P=\displaystyle \log_{\phi(1729)}\left(\frac{1}{\phi(1729)^4}\right)=-4$

$\boxed{\therefore |P|=4}$