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Recall that the primorial $P_n$ is the product of the first $n$ primes.

We say that a number $m$ is good if and only if for every $N$ the fraction of numbers in the range $\{1,2,3,\dots, N\}$ that are relatively prime to $m$ is greater than or equal to $\varphi(m)/m$.

Determine all good primorials.


Clearly it suffices to check for $N\in \{1,2,3,\dots, m\}$

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  • $\begingroup$ You are right, I am so sorry. It should be ok now. $\endgroup$ – Jorge Fernández Hidalgo Dec 25 '16 at 20:24
  • $\begingroup$ Your thoughts on this problem are...? $\endgroup$ – barak manos Dec 25 '16 at 20:33
  • $\begingroup$ 210,2310 and 30030 are not good. $\endgroup$ – Jorge Fernández Hidalgo Dec 25 '16 at 20:53
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Let $P_n$ be the $n$-th primorial and $p_n$ the $n$-th prime. Then $$\frac{\varphi(P_n)}{P_n}=\prod_{i=1}^n(1-p_i^{-1}),$$ and the only integer in the set $\{1,\ldots,p_{n+1}-1\}$ coprime to $P_n$ is $1$, so the fraction of integers coprime to $P_n$ in this range is $\frac{1}{p_{n+1}-1}$. I'll leave it to you to verify that $$\prod_{i=1}^n(1-p_i^{-1})>\frac{1}{p_{n+1}-1},$$ for every $n>1$, so the only good primorial is $P_1=2$.

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