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Bai lu zhou Academy Problem

Source: 2022 China Southeast Grade 10 P3 and 11 P4

August 2, 2022
algebranumber theory

Problem Statement

If xix_i is an integer greater than 1, let f(xi)f(x_i) be the greatest prime factor of xi,xi+1=xi−f(xi)x_i,x_{i+1} =x_i-f(x_i) (i≥0i\ge 0 and i is an integer). (1) Prove that for any integer x0x_0 greater than 1, there exists a natural numberk(x0)k(x_0), such that xk(x0)+1=0x_{k(x_0)+1}=0 Grade 10: (2) Let V(x0)V_{(x_0)} be the number of different numbers in f(x0),f(x1),⋯ ,f(xk(x0))f(x_0),f(x_1),\cdots,f(x_{k(x_0)}). Find the largest number in V(2),V(3),⋯ ,V(781)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(x0)V_{(x_0)} be the number of different numbers in f(x0),f(x1),⋯ ,f(xk(x0))f(x_0),f(x_1),\cdots,f(x_{k(x_0)}). Find the largest number in V(2),V(3),⋯ ,V(2022)V(2),V(3),\cdots,V(2022) and give reasons.
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