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max m, x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)/{m!} never multiple of 7

Source: 2022 NZMO - New Zealand Maths Olympiad Round 2 p5

October 8, 2022
number theorymultipledividesdivisiblerecurrence relation

Problem Statement

The sequence x1,x2,x3,...x_1, x_2, x_3, . . . is defined by x1=2022x_1 = 2022 and xn+1=7xn+5x_{n+1}= 7x_n + 5 for all positive integers nn. Determine the maximum positive integer mm such that xn(xn−1)(xn−2)...(xn−m+1)m!\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!} is never a multiple of 77 for any positive integer nn.
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