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Prove that for every prime number $p$ the number $x_p-1$ is divisible by $3p.$

Source: Moldova TST 2022

April 1, 2022
number theory

Problem Statement

Let (xn)n≥1(x_n)_{n\geq1} be a sequence that verifies: x_1=1,   x_2=7,   x_{n+1}=x_n+3x_{n-1}, \forall n \geq 2. Prove that for every prime number pp the number xp−1x_p-1 is divisible by 3p.3p.
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