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x_n = ax_{n-1} + b, m | n => x_m | x_n

Source: Czech-Polish-Slovak Match 2014 day 1 P2

October 6, 2017
number theoryarithmetic sequence

Problem Statement

For the positive integers a,b,x1a, b, x_1 we construct the sequence of numbers (xn)n=1∞(x_n)_{n=1}^{\infty} such that xn=axn−1+bx_n = ax_{n-1} + b for each n≥2n \ge 2. Specify the conditions for the given numbers a,ba, b and x1x_1 which are necessary and sufficient for all indexes m,nm, n to apply the implication m∣n⇒xm∣xnm | n \Rightarrow x_m | x_n.
(Jaromír Šimša)
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