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December 29th, 2018, 12:58 AM   #11
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The first thing you'd need to prove is that $\varphi(n)$ is not $<5$ infinitely many times. So you'd need to prove there actually IS an N such that for $n\geq N$, $\varphi(n)\geq 5$.

Use the prime decompositions for that, and the formula of $\varphi$ on such decompositions.
When proving this, you'll be able to constructively find a value of $N$. Then it is just a matter of checking all values from $1$ to $N$.
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December 29th, 2018, 10:38 PM   #12
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Here's a start: For any positive integer $n$, if $n$ is divisible by a prime factor $p \ge 7$, then $\phi(n)$ contains $p-1 \ge 6$ as a factor. So any number $n$ with $\phi(n) < 5$ can only have 2, 3, and 5 as prime factors.
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