255's blog

By 255, history, 7 weeks ago, In English

Hello Codeforces, recently, there was a contest on CSAcademy named "FiiCode 2021 Round 2" and there was a task "Clown Fiesta", in a editorial for the task, there were lines such like:

If we take a sequence, where $$$a_0 = m$$$, $$$a_1 = \varphi (a_0)$$$, $$$a_2 = \varphi (a_1)$$$, $$$...$$$, $$$a_{k-1} = \varphi (a_{k-2}) \neq 1$$$, $$$a_k = \varphi (a_{k-1}) = 1$$$ and $$$m$$$ is prime number less than $$$10^9$$$, then $$$k$$$ will be not greater than $$$log_2(m)$$$.

Can someone prove it?

P.S. link to problem.

 
 
 
 
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7 weeks ago, # |
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A not formal proof.

If x is odd, phi(x) is even

If x is even, phi(x) <= x/2

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    7 weeks ago, # ^ |
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    got it, thanks)

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    7 weeks ago, # ^ |
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    This proofs, that $$$k \leq 2 * log_2(m)$$$. Actually we can proof, that $$$k \leq log_2(m) + 1$$$, because $$$\varphi(x)$$$ is even for all $$$x > 2$$$.

    Proof: If $$$gcd(n, x) = 1$$$, then $$$gcd(n, n - x) = 1$$$ too, and if $$$n$$$ is even and $$$n > 2$$$, then $$$gcd(n, \frac{n}{2}) \neq 1$$$, so all numbers coprime with $$$n$$$ are splited in pairs.