If $x_i$ is an integer greater than 1, let $f(x_i)$ be the greatest prime factor of $x_i,x_{i+1} =x_i-f(x_i)$ ($i\ge 0$ and i is an integer). (1) Prove that for any integer $x_0$ greater than 1, there exists a natural number$k(x_0)$, such that $x_{k(x_0)+1}=0$ Grade 10: (2) Let $V_{(x_0)}$ be the number of different numbers in $f(x_0),f(x_1),\cdots,f(x_{k(x_0)})$. Find the largest number in $V(2),V(3),\cdots,V(781)$ and give reasons. Note: Bai Lu Zhou Academy was founded in 1241 and has a history of 781 years. Grade 11: (2) Let $V_{(x_0)}$ be the number of different numbers in $f(x_0),f(x_1),\cdots,f(x_{k(x_0)})$. Find the largest number in $V(2),V(3),\cdots,V(2022)$ and give reasons.
Problem
Source: 2022 China Southeast Grade 10 P3 and 11 P4
Tags: algebra, number theory
02.08.2022 08:51
@above This is not Grade 11 P3. Grade 11 P4 asks what is the largest number in V(2) to V(2022).
02.08.2022 09:46
Is this a typing mistake? I don't see $x$, I only see $x_i$ here.
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02.08.2022 09:49
Thanks @2above, @below
02.08.2022 09:49
Also, I don't get this line, is the last $x_{k(x_0)}$ or $f(x_{k(x_0)})$
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04.08.2022 22:00
Hm, this problem seems hard, any hint? (without solution by computer program of course)
05.08.2022 05:36
Let $p_i$ be the largest prime factor of $x_i$, then $x_i$ strictly decreases while $p_i$ monotonically increases, solving (i). To solve (ii), we just need to find the maximum number of times $p_i$ increases. This should probably be a big bash.